Non-inductive, not combinatorial proof of $sum_{i mathop = 0}^n binom n i^2 = binom {2 n} n$
I've seen the identity $displaystyle sum_{i mathop = 0}^n binom n i^2 = binom {2 n} n$ used here recently.
I checked for proofs here http://www.proofwiki.org/wiki/Sum_of_Squares_of_Binomial_Coefficients
I couldn't have figured out the combinatorial proof by myself, and the inductive proof assumes you already know the answer...
So my question is : do you know how to prove directly through computation that $displaystyle sum_{i mathop = 0}^n binom n i^2 = binom {2 n} n$ ?
combinatorics binomial-coefficients
add a comment |
I've seen the identity $displaystyle sum_{i mathop = 0}^n binom n i^2 = binom {2 n} n$ used here recently.
I checked for proofs here http://www.proofwiki.org/wiki/Sum_of_Squares_of_Binomial_Coefficients
I couldn't have figured out the combinatorial proof by myself, and the inductive proof assumes you already know the answer...
So my question is : do you know how to prove directly through computation that $displaystyle sum_{i mathop = 0}^n binom n i^2 = binom {2 n} n$ ?
combinatorics binomial-coefficients
add a comment |
I've seen the identity $displaystyle sum_{i mathop = 0}^n binom n i^2 = binom {2 n} n$ used here recently.
I checked for proofs here http://www.proofwiki.org/wiki/Sum_of_Squares_of_Binomial_Coefficients
I couldn't have figured out the combinatorial proof by myself, and the inductive proof assumes you already know the answer...
So my question is : do you know how to prove directly through computation that $displaystyle sum_{i mathop = 0}^n binom n i^2 = binom {2 n} n$ ?
combinatorics binomial-coefficients
I've seen the identity $displaystyle sum_{i mathop = 0}^n binom n i^2 = binom {2 n} n$ used here recently.
I checked for proofs here http://www.proofwiki.org/wiki/Sum_of_Squares_of_Binomial_Coefficients
I couldn't have figured out the combinatorial proof by myself, and the inductive proof assumes you already know the answer...
So my question is : do you know how to prove directly through computation that $displaystyle sum_{i mathop = 0}^n binom n i^2 = binom {2 n} n$ ?
combinatorics binomial-coefficients
combinatorics binomial-coefficients
edited May 13 '14 at 15:44
WLOG
7,21932258
7,21932258
asked May 13 '14 at 15:13
Gabriel Romon
17.8k53284
17.8k53284
add a comment |
add a comment |
5 Answers
5
active
oldest
votes
Consider the identity $(1+x)^{2n}=(1+x)^ncdot(1+x)^n$. By the binomial theorem we have $displaystyle(1+x)^n=sum_{k=0}^n{nchoose k} x^k$, so multiplying out we compute the right hand side as $displaystyle(1+x)^ncdot(1+x)^n = sum_{k=0}^{2n}left(sum_{j=0}^k{nchoose j}{nchoose k-j}x^kright)$. But the LHS is just $displaystyle(1+x)^{2n} = sum_{k=0}^{2n}{2nchoose k}x^k$; equating coefficients of $x^n$ we get $displaystyle{2nchoose n}=sum_{j=0}^n {nchoose j}{nchoose n-j}$. Finally, using the identity ${nchoose j}={nchoose n-j}$ gives the desired result.
(and use the identity that $nchoose i$=$nchoose n-i$)
– Steven Stadnicki
May 13 '14 at 15:16
(Using the binomial formula, of course.)
– Arthur
May 13 '14 at 15:16
@StevenStadnicki, With the current version, we don't need that Identity
– lab bhattacharjee
May 13 '14 at 15:26
@labbhattacharjee I think you do need it after Cauchy product.
– Gabriel Romon
May 13 '14 at 15:29
I fail to see how this is a proof.
– Superbus
May 13 '14 at 15:55
|
show 4 more comments
lab bhattacharjee has given the proof, but it is worth pointing out that this formula is simply an application of the Vandermonde convolution, which says that
$$sum_{j=0}^k binom{n}{j}binom{m}{k-j} = binom{n+m}{k}.$$
Setting $n=m=k$ and noting that $binom{n}{n-j}=binom{n}{j}$ gives the result.
EDIT: By the way, for a slightly non-standard (but purely technical, as you require) proof of the Vandermonde convolution, let $X_1,cdots,X_{n+m}$ be independent Bernoulli distributed RV's with parameter $p=1/2$. Then, $S_{n+m}=sum_{j=0}^{n+m}X_j$ has binomial distribution $P[S_{n+m}=k]=binom{n+m}{k}(frac{1}{2})^{n+m}$. Applying the ordinary discrete convolution formula for probability distributions yields
$$(1/2)^{n+m}binom{n+m}{k}=P[S_{n+m}=k]=sum_{j=0}^k P[S_n=j]P[S_m=k-j]=(1/2)^{n+m}sum_{j=0}^kbinom{n}{j}binom{m}{k-j}.$$
add a comment |
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${ttcolor{#f00}{mbox{This method is a "direct calculation" which is very}}}$
${ttcolor{#f00}{mbox{ convenient when we don't know the answer and, in addition,}}}$
${ttcolor{#f00}{mbox{ we don't have to guess combinations of Newton binomials.}}}$
It's based in the identity:
$$
{m choose s}
=oint_{verts{z} = 1}{pars{1 + z}^{m} over z^{s + 1}},{dd z over 2piic}
$$
begin{align}
&bbox[10px,#ffd]{sum_{k = 0}^{n}{n choose k}^{2}} =
sum_{k = 0}^{n}{n choose k}
oint_{verts{z} = 1}{pars{1 + z}^{n} over
z^{k + 1}},{dd z over 2piic}
\[5mm] = &
oint_{verts{z} = 1}{pars{1 + z}^{n} over z}
sum_{k = 0}^{n}{n choose k}pars{1 over z}^{k}
,{dd z over 2piic}
\[3mm]&=oint_{verts{z} = 1}{pars{1 + z}^{n} over z}
pars{1 + {1 over z}}^{n},{dd z over 2piic}
=oint_{verts{z} = 1}{pars{1 + z}^{2n} over z^{n + 1}},{dd z over 2piic}
\[5mm] = &
bbox[10px,border:1px groove navy]{2n choose n}
end{align}
very cool, any tip on how to show $$sum_{k=0}^{n}binom{n}{k}left(frac{1}{z}right)^k = left(1+frac{1}{z}right)^n?$$
– chs21259
Jul 8 '14 at 2:12
@chs21259 That follows from the binomial theorem.
– Dan Z
Jul 8 '14 at 2:39
@DanZollers oh wow, duh, thanks Dan
– chs21259
Jul 8 '14 at 2:40
add a comment |
You're after a sum of a hypergeometric term. There are general techniques for finding closed forms or proving that they don't exist. See e.g. the book A = B, or MathWorld's description of Sister Celine's method, Gosper's algorithm, and Zeilberger's algorithm.
add a comment |
Note that we can divide the $2n$ objects in $2$ groups including $n$ elements each.
Then to choose $n$ elements out of $2n$ we can choose $0$ objects in the first group and $n$ in the second, or $1$ in the first and $n-1$ in the second and so on, i.e $$binom{2n}{n} = sum_{i = 0}^{2n}binom{n}{i}binom{n}{n-i} =sum_{i = 0}^{2n}{binom{n}{i}}^2 $$
3
This is a combinatorial proof, which is contrary to what OP wanted.
– Batman
May 13 '14 at 17:43
add a comment |
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5 Answers
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5 Answers
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Consider the identity $(1+x)^{2n}=(1+x)^ncdot(1+x)^n$. By the binomial theorem we have $displaystyle(1+x)^n=sum_{k=0}^n{nchoose k} x^k$, so multiplying out we compute the right hand side as $displaystyle(1+x)^ncdot(1+x)^n = sum_{k=0}^{2n}left(sum_{j=0}^k{nchoose j}{nchoose k-j}x^kright)$. But the LHS is just $displaystyle(1+x)^{2n} = sum_{k=0}^{2n}{2nchoose k}x^k$; equating coefficients of $x^n$ we get $displaystyle{2nchoose n}=sum_{j=0}^n {nchoose j}{nchoose n-j}$. Finally, using the identity ${nchoose j}={nchoose n-j}$ gives the desired result.
(and use the identity that $nchoose i$=$nchoose n-i$)
– Steven Stadnicki
May 13 '14 at 15:16
(Using the binomial formula, of course.)
– Arthur
May 13 '14 at 15:16
@StevenStadnicki, With the current version, we don't need that Identity
– lab bhattacharjee
May 13 '14 at 15:26
@labbhattacharjee I think you do need it after Cauchy product.
– Gabriel Romon
May 13 '14 at 15:29
I fail to see how this is a proof.
– Superbus
May 13 '14 at 15:55
|
show 4 more comments
Consider the identity $(1+x)^{2n}=(1+x)^ncdot(1+x)^n$. By the binomial theorem we have $displaystyle(1+x)^n=sum_{k=0}^n{nchoose k} x^k$, so multiplying out we compute the right hand side as $displaystyle(1+x)^ncdot(1+x)^n = sum_{k=0}^{2n}left(sum_{j=0}^k{nchoose j}{nchoose k-j}x^kright)$. But the LHS is just $displaystyle(1+x)^{2n} = sum_{k=0}^{2n}{2nchoose k}x^k$; equating coefficients of $x^n$ we get $displaystyle{2nchoose n}=sum_{j=0}^n {nchoose j}{nchoose n-j}$. Finally, using the identity ${nchoose j}={nchoose n-j}$ gives the desired result.
(and use the identity that $nchoose i$=$nchoose n-i$)
– Steven Stadnicki
May 13 '14 at 15:16
(Using the binomial formula, of course.)
– Arthur
May 13 '14 at 15:16
@StevenStadnicki, With the current version, we don't need that Identity
– lab bhattacharjee
May 13 '14 at 15:26
@labbhattacharjee I think you do need it after Cauchy product.
– Gabriel Romon
May 13 '14 at 15:29
I fail to see how this is a proof.
– Superbus
May 13 '14 at 15:55
|
show 4 more comments
Consider the identity $(1+x)^{2n}=(1+x)^ncdot(1+x)^n$. By the binomial theorem we have $displaystyle(1+x)^n=sum_{k=0}^n{nchoose k} x^k$, so multiplying out we compute the right hand side as $displaystyle(1+x)^ncdot(1+x)^n = sum_{k=0}^{2n}left(sum_{j=0}^k{nchoose j}{nchoose k-j}x^kright)$. But the LHS is just $displaystyle(1+x)^{2n} = sum_{k=0}^{2n}{2nchoose k}x^k$; equating coefficients of $x^n$ we get $displaystyle{2nchoose n}=sum_{j=0}^n {nchoose j}{nchoose n-j}$. Finally, using the identity ${nchoose j}={nchoose n-j}$ gives the desired result.
Consider the identity $(1+x)^{2n}=(1+x)^ncdot(1+x)^n$. By the binomial theorem we have $displaystyle(1+x)^n=sum_{k=0}^n{nchoose k} x^k$, so multiplying out we compute the right hand side as $displaystyle(1+x)^ncdot(1+x)^n = sum_{k=0}^{2n}left(sum_{j=0}^k{nchoose j}{nchoose k-j}x^kright)$. But the LHS is just $displaystyle(1+x)^{2n} = sum_{k=0}^{2n}{2nchoose k}x^k$; equating coefficients of $x^n$ we get $displaystyle{2nchoose n}=sum_{j=0}^n {nchoose j}{nchoose n-j}$. Finally, using the identity ${nchoose j}={nchoose n-j}$ gives the desired result.
edited May 13 '14 at 17:41
Steven Stadnicki
41k867122
41k867122
answered May 13 '14 at 15:15
lab bhattacharjee
223k15156274
223k15156274
(and use the identity that $nchoose i$=$nchoose n-i$)
– Steven Stadnicki
May 13 '14 at 15:16
(Using the binomial formula, of course.)
– Arthur
May 13 '14 at 15:16
@StevenStadnicki, With the current version, we don't need that Identity
– lab bhattacharjee
May 13 '14 at 15:26
@labbhattacharjee I think you do need it after Cauchy product.
– Gabriel Romon
May 13 '14 at 15:29
I fail to see how this is a proof.
– Superbus
May 13 '14 at 15:55
|
show 4 more comments
(and use the identity that $nchoose i$=$nchoose n-i$)
– Steven Stadnicki
May 13 '14 at 15:16
(Using the binomial formula, of course.)
– Arthur
May 13 '14 at 15:16
@StevenStadnicki, With the current version, we don't need that Identity
– lab bhattacharjee
May 13 '14 at 15:26
@labbhattacharjee I think you do need it after Cauchy product.
– Gabriel Romon
May 13 '14 at 15:29
I fail to see how this is a proof.
– Superbus
May 13 '14 at 15:55
(and use the identity that $nchoose i$=$nchoose n-i$)
– Steven Stadnicki
May 13 '14 at 15:16
(and use the identity that $nchoose i$=$nchoose n-i$)
– Steven Stadnicki
May 13 '14 at 15:16
(Using the binomial formula, of course.)
– Arthur
May 13 '14 at 15:16
(Using the binomial formula, of course.)
– Arthur
May 13 '14 at 15:16
@StevenStadnicki, With the current version, we don't need that Identity
– lab bhattacharjee
May 13 '14 at 15:26
@StevenStadnicki, With the current version, we don't need that Identity
– lab bhattacharjee
May 13 '14 at 15:26
@labbhattacharjee I think you do need it after Cauchy product.
– Gabriel Romon
May 13 '14 at 15:29
@labbhattacharjee I think you do need it after Cauchy product.
– Gabriel Romon
May 13 '14 at 15:29
I fail to see how this is a proof.
– Superbus
May 13 '14 at 15:55
I fail to see how this is a proof.
– Superbus
May 13 '14 at 15:55
|
show 4 more comments
lab bhattacharjee has given the proof, but it is worth pointing out that this formula is simply an application of the Vandermonde convolution, which says that
$$sum_{j=0}^k binom{n}{j}binom{m}{k-j} = binom{n+m}{k}.$$
Setting $n=m=k$ and noting that $binom{n}{n-j}=binom{n}{j}$ gives the result.
EDIT: By the way, for a slightly non-standard (but purely technical, as you require) proof of the Vandermonde convolution, let $X_1,cdots,X_{n+m}$ be independent Bernoulli distributed RV's with parameter $p=1/2$. Then, $S_{n+m}=sum_{j=0}^{n+m}X_j$ has binomial distribution $P[S_{n+m}=k]=binom{n+m}{k}(frac{1}{2})^{n+m}$. Applying the ordinary discrete convolution formula for probability distributions yields
$$(1/2)^{n+m}binom{n+m}{k}=P[S_{n+m}=k]=sum_{j=0}^k P[S_n=j]P[S_m=k-j]=(1/2)^{n+m}sum_{j=0}^kbinom{n}{j}binom{m}{k-j}.$$
add a comment |
lab bhattacharjee has given the proof, but it is worth pointing out that this formula is simply an application of the Vandermonde convolution, which says that
$$sum_{j=0}^k binom{n}{j}binom{m}{k-j} = binom{n+m}{k}.$$
Setting $n=m=k$ and noting that $binom{n}{n-j}=binom{n}{j}$ gives the result.
EDIT: By the way, for a slightly non-standard (but purely technical, as you require) proof of the Vandermonde convolution, let $X_1,cdots,X_{n+m}$ be independent Bernoulli distributed RV's with parameter $p=1/2$. Then, $S_{n+m}=sum_{j=0}^{n+m}X_j$ has binomial distribution $P[S_{n+m}=k]=binom{n+m}{k}(frac{1}{2})^{n+m}$. Applying the ordinary discrete convolution formula for probability distributions yields
$$(1/2)^{n+m}binom{n+m}{k}=P[S_{n+m}=k]=sum_{j=0}^k P[S_n=j]P[S_m=k-j]=(1/2)^{n+m}sum_{j=0}^kbinom{n}{j}binom{m}{k-j}.$$
add a comment |
lab bhattacharjee has given the proof, but it is worth pointing out that this formula is simply an application of the Vandermonde convolution, which says that
$$sum_{j=0}^k binom{n}{j}binom{m}{k-j} = binom{n+m}{k}.$$
Setting $n=m=k$ and noting that $binom{n}{n-j}=binom{n}{j}$ gives the result.
EDIT: By the way, for a slightly non-standard (but purely technical, as you require) proof of the Vandermonde convolution, let $X_1,cdots,X_{n+m}$ be independent Bernoulli distributed RV's with parameter $p=1/2$. Then, $S_{n+m}=sum_{j=0}^{n+m}X_j$ has binomial distribution $P[S_{n+m}=k]=binom{n+m}{k}(frac{1}{2})^{n+m}$. Applying the ordinary discrete convolution formula for probability distributions yields
$$(1/2)^{n+m}binom{n+m}{k}=P[S_{n+m}=k]=sum_{j=0}^k P[S_n=j]P[S_m=k-j]=(1/2)^{n+m}sum_{j=0}^kbinom{n}{j}binom{m}{k-j}.$$
lab bhattacharjee has given the proof, but it is worth pointing out that this formula is simply an application of the Vandermonde convolution, which says that
$$sum_{j=0}^k binom{n}{j}binom{m}{k-j} = binom{n+m}{k}.$$
Setting $n=m=k$ and noting that $binom{n}{n-j}=binom{n}{j}$ gives the result.
EDIT: By the way, for a slightly non-standard (but purely technical, as you require) proof of the Vandermonde convolution, let $X_1,cdots,X_{n+m}$ be independent Bernoulli distributed RV's with parameter $p=1/2$. Then, $S_{n+m}=sum_{j=0}^{n+m}X_j$ has binomial distribution $P[S_{n+m}=k]=binom{n+m}{k}(frac{1}{2})^{n+m}$. Applying the ordinary discrete convolution formula for probability distributions yields
$$(1/2)^{n+m}binom{n+m}{k}=P[S_{n+m}=k]=sum_{j=0}^k P[S_n=j]P[S_m=k-j]=(1/2)^{n+m}sum_{j=0}^kbinom{n}{j}binom{m}{k-j}.$$
edited May 13 '14 at 16:06
answered May 13 '14 at 15:29
mathse
2,012513
2,012513
add a comment |
add a comment |
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${ttcolor{#f00}{mbox{This method is a "direct calculation" which is very}}}$
${ttcolor{#f00}{mbox{ convenient when we don't know the answer and, in addition,}}}$
${ttcolor{#f00}{mbox{ we don't have to guess combinations of Newton binomials.}}}$
It's based in the identity:
$$
{m choose s}
=oint_{verts{z} = 1}{pars{1 + z}^{m} over z^{s + 1}},{dd z over 2piic}
$$
begin{align}
&bbox[10px,#ffd]{sum_{k = 0}^{n}{n choose k}^{2}} =
sum_{k = 0}^{n}{n choose k}
oint_{verts{z} = 1}{pars{1 + z}^{n} over
z^{k + 1}},{dd z over 2piic}
\[5mm] = &
oint_{verts{z} = 1}{pars{1 + z}^{n} over z}
sum_{k = 0}^{n}{n choose k}pars{1 over z}^{k}
,{dd z over 2piic}
\[3mm]&=oint_{verts{z} = 1}{pars{1 + z}^{n} over z}
pars{1 + {1 over z}}^{n},{dd z over 2piic}
=oint_{verts{z} = 1}{pars{1 + z}^{2n} over z^{n + 1}},{dd z over 2piic}
\[5mm] = &
bbox[10px,border:1px groove navy]{2n choose n}
end{align}
very cool, any tip on how to show $$sum_{k=0}^{n}binom{n}{k}left(frac{1}{z}right)^k = left(1+frac{1}{z}right)^n?$$
– chs21259
Jul 8 '14 at 2:12
@chs21259 That follows from the binomial theorem.
– Dan Z
Jul 8 '14 at 2:39
@DanZollers oh wow, duh, thanks Dan
– chs21259
Jul 8 '14 at 2:40
add a comment |
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newcommand{wt}[1]{widetilde{#1}}$
${ttcolor{#f00}{mbox{This method is a "direct calculation" which is very}}}$
${ttcolor{#f00}{mbox{ convenient when we don't know the answer and, in addition,}}}$
${ttcolor{#f00}{mbox{ we don't have to guess combinations of Newton binomials.}}}$
It's based in the identity:
$$
{m choose s}
=oint_{verts{z} = 1}{pars{1 + z}^{m} over z^{s + 1}},{dd z over 2piic}
$$
begin{align}
&bbox[10px,#ffd]{sum_{k = 0}^{n}{n choose k}^{2}} =
sum_{k = 0}^{n}{n choose k}
oint_{verts{z} = 1}{pars{1 + z}^{n} over
z^{k + 1}},{dd z over 2piic}
\[5mm] = &
oint_{verts{z} = 1}{pars{1 + z}^{n} over z}
sum_{k = 0}^{n}{n choose k}pars{1 over z}^{k}
,{dd z over 2piic}
\[3mm]&=oint_{verts{z} = 1}{pars{1 + z}^{n} over z}
pars{1 + {1 over z}}^{n},{dd z over 2piic}
=oint_{verts{z} = 1}{pars{1 + z}^{2n} over z^{n + 1}},{dd z over 2piic}
\[5mm] = &
bbox[10px,border:1px groove navy]{2n choose n}
end{align}
very cool, any tip on how to show $$sum_{k=0}^{n}binom{n}{k}left(frac{1}{z}right)^k = left(1+frac{1}{z}right)^n?$$
– chs21259
Jul 8 '14 at 2:12
@chs21259 That follows from the binomial theorem.
– Dan Z
Jul 8 '14 at 2:39
@DanZollers oh wow, duh, thanks Dan
– chs21259
Jul 8 '14 at 2:40
add a comment |
$newcommand{+}{^{dagger}}
newcommand{angles}[1]{leftlangle, #1 ,rightrangle}
newcommand{braces}[1]{leftlbrace, #1 ,rightrbrace}
newcommand{bracks}[1]{leftlbrack, #1 ,rightrbrack}
newcommand{ceil}[1]{,leftlceil, #1 ,rightrceil,}
newcommand{dd}{{rm d}}
newcommand{down}{downarrow}
newcommand{ds}[1]{displaystyle{#1}}
newcommand{expo}[1]{,{rm e}^{#1},}
newcommand{fermi}{,{rm f}}
newcommand{floor}[1]{,leftlfloor #1 rightrfloor,}
newcommand{half}{{1 over 2}}
newcommand{ic}{{rm i}}
newcommand{iff}{Longleftrightarrow}
newcommand{imp}{Longrightarrow}
newcommand{isdiv}{,left.rightvert,}
newcommand{ket}[1]{leftvert #1rightrangle}
newcommand{ol}[1]{overline{#1}}
newcommand{pars}[1]{left(, #1 ,right)}
newcommand{partiald}[3]{frac{partial^{#1} #2}{partial #3^{#1}}}
newcommand{pp}{{cal P}}
newcommand{root}[2]{,sqrt[#1]{vphantom{large A},#2,},}
newcommand{sech}{,{rm sech}}
newcommand{sgn}{,{rm sgn}}
newcommand{totald}[3]{frac{{rm d}^{#1} #2}{{rm d} #3^{#1}}}
newcommand{ul}[1]{underline{#1}}
newcommand{verts}[1]{leftvert, #1 ,rightvert}
newcommand{wt}[1]{widetilde{#1}}$
${ttcolor{#f00}{mbox{This method is a "direct calculation" which is very}}}$
${ttcolor{#f00}{mbox{ convenient when we don't know the answer and, in addition,}}}$
${ttcolor{#f00}{mbox{ we don't have to guess combinations of Newton binomials.}}}$
It's based in the identity:
$$
{m choose s}
=oint_{verts{z} = 1}{pars{1 + z}^{m} over z^{s + 1}},{dd z over 2piic}
$$
begin{align}
&bbox[10px,#ffd]{sum_{k = 0}^{n}{n choose k}^{2}} =
sum_{k = 0}^{n}{n choose k}
oint_{verts{z} = 1}{pars{1 + z}^{n} over
z^{k + 1}},{dd z over 2piic}
\[5mm] = &
oint_{verts{z} = 1}{pars{1 + z}^{n} over z}
sum_{k = 0}^{n}{n choose k}pars{1 over z}^{k}
,{dd z over 2piic}
\[3mm]&=oint_{verts{z} = 1}{pars{1 + z}^{n} over z}
pars{1 + {1 over z}}^{n},{dd z over 2piic}
=oint_{verts{z} = 1}{pars{1 + z}^{2n} over z^{n + 1}},{dd z over 2piic}
\[5mm] = &
bbox[10px,border:1px groove navy]{2n choose n}
end{align}
$newcommand{+}{^{dagger}}
newcommand{angles}[1]{leftlangle, #1 ,rightrangle}
newcommand{braces}[1]{leftlbrace, #1 ,rightrbrace}
newcommand{bracks}[1]{leftlbrack, #1 ,rightrbrack}
newcommand{ceil}[1]{,leftlceil, #1 ,rightrceil,}
newcommand{dd}{{rm d}}
newcommand{down}{downarrow}
newcommand{ds}[1]{displaystyle{#1}}
newcommand{expo}[1]{,{rm e}^{#1},}
newcommand{fermi}{,{rm f}}
newcommand{floor}[1]{,leftlfloor #1 rightrfloor,}
newcommand{half}{{1 over 2}}
newcommand{ic}{{rm i}}
newcommand{iff}{Longleftrightarrow}
newcommand{imp}{Longrightarrow}
newcommand{isdiv}{,left.rightvert,}
newcommand{ket}[1]{leftvert #1rightrangle}
newcommand{ol}[1]{overline{#1}}
newcommand{pars}[1]{left(, #1 ,right)}
newcommand{partiald}[3]{frac{partial^{#1} #2}{partial #3^{#1}}}
newcommand{pp}{{cal P}}
newcommand{root}[2]{,sqrt[#1]{vphantom{large A},#2,},}
newcommand{sech}{,{rm sech}}
newcommand{sgn}{,{rm sgn}}
newcommand{totald}[3]{frac{{rm d}^{#1} #2}{{rm d} #3^{#1}}}
newcommand{ul}[1]{underline{#1}}
newcommand{verts}[1]{leftvert, #1 ,rightvert}
newcommand{wt}[1]{widetilde{#1}}$
${ttcolor{#f00}{mbox{This method is a "direct calculation" which is very}}}$
${ttcolor{#f00}{mbox{ convenient when we don't know the answer and, in addition,}}}$
${ttcolor{#f00}{mbox{ we don't have to guess combinations of Newton binomials.}}}$
It's based in the identity:
$$
{m choose s}
=oint_{verts{z} = 1}{pars{1 + z}^{m} over z^{s + 1}},{dd z over 2piic}
$$
begin{align}
&bbox[10px,#ffd]{sum_{k = 0}^{n}{n choose k}^{2}} =
sum_{k = 0}^{n}{n choose k}
oint_{verts{z} = 1}{pars{1 + z}^{n} over
z^{k + 1}},{dd z over 2piic}
\[5mm] = &
oint_{verts{z} = 1}{pars{1 + z}^{n} over z}
sum_{k = 0}^{n}{n choose k}pars{1 over z}^{k}
,{dd z over 2piic}
\[3mm]&=oint_{verts{z} = 1}{pars{1 + z}^{n} over z}
pars{1 + {1 over z}}^{n},{dd z over 2piic}
=oint_{verts{z} = 1}{pars{1 + z}^{2n} over z^{n + 1}},{dd z over 2piic}
\[5mm] = &
bbox[10px,border:1px groove navy]{2n choose n}
end{align}
edited Dec 4 '18 at 20:02
answered Jul 8 '14 at 1:19
Felix Marin
67.2k7107141
67.2k7107141
very cool, any tip on how to show $$sum_{k=0}^{n}binom{n}{k}left(frac{1}{z}right)^k = left(1+frac{1}{z}right)^n?$$
– chs21259
Jul 8 '14 at 2:12
@chs21259 That follows from the binomial theorem.
– Dan Z
Jul 8 '14 at 2:39
@DanZollers oh wow, duh, thanks Dan
– chs21259
Jul 8 '14 at 2:40
add a comment |
very cool, any tip on how to show $$sum_{k=0}^{n}binom{n}{k}left(frac{1}{z}right)^k = left(1+frac{1}{z}right)^n?$$
– chs21259
Jul 8 '14 at 2:12
@chs21259 That follows from the binomial theorem.
– Dan Z
Jul 8 '14 at 2:39
@DanZollers oh wow, duh, thanks Dan
– chs21259
Jul 8 '14 at 2:40
very cool, any tip on how to show $$sum_{k=0}^{n}binom{n}{k}left(frac{1}{z}right)^k = left(1+frac{1}{z}right)^n?$$
– chs21259
Jul 8 '14 at 2:12
very cool, any tip on how to show $$sum_{k=0}^{n}binom{n}{k}left(frac{1}{z}right)^k = left(1+frac{1}{z}right)^n?$$
– chs21259
Jul 8 '14 at 2:12
@chs21259 That follows from the binomial theorem.
– Dan Z
Jul 8 '14 at 2:39
@chs21259 That follows from the binomial theorem.
– Dan Z
Jul 8 '14 at 2:39
@DanZollers oh wow, duh, thanks Dan
– chs21259
Jul 8 '14 at 2:40
@DanZollers oh wow, duh, thanks Dan
– chs21259
Jul 8 '14 at 2:40
add a comment |
You're after a sum of a hypergeometric term. There are general techniques for finding closed forms or proving that they don't exist. See e.g. the book A = B, or MathWorld's description of Sister Celine's method, Gosper's algorithm, and Zeilberger's algorithm.
add a comment |
You're after a sum of a hypergeometric term. There are general techniques for finding closed forms or proving that they don't exist. See e.g. the book A = B, or MathWorld's description of Sister Celine's method, Gosper's algorithm, and Zeilberger's algorithm.
add a comment |
You're after a sum of a hypergeometric term. There are general techniques for finding closed forms or proving that they don't exist. See e.g. the book A = B, or MathWorld's description of Sister Celine's method, Gosper's algorithm, and Zeilberger's algorithm.
You're after a sum of a hypergeometric term. There are general techniques for finding closed forms or proving that they don't exist. See e.g. the book A = B, or MathWorld's description of Sister Celine's method, Gosper's algorithm, and Zeilberger's algorithm.
answered May 13 '14 at 15:35
Peter Taylor
8,70812241
8,70812241
add a comment |
add a comment |
Note that we can divide the $2n$ objects in $2$ groups including $n$ elements each.
Then to choose $n$ elements out of $2n$ we can choose $0$ objects in the first group and $n$ in the second, or $1$ in the first and $n-1$ in the second and so on, i.e $$binom{2n}{n} = sum_{i = 0}^{2n}binom{n}{i}binom{n}{n-i} =sum_{i = 0}^{2n}{binom{n}{i}}^2 $$
3
This is a combinatorial proof, which is contrary to what OP wanted.
– Batman
May 13 '14 at 17:43
add a comment |
Note that we can divide the $2n$ objects in $2$ groups including $n$ elements each.
Then to choose $n$ elements out of $2n$ we can choose $0$ objects in the first group and $n$ in the second, or $1$ in the first and $n-1$ in the second and so on, i.e $$binom{2n}{n} = sum_{i = 0}^{2n}binom{n}{i}binom{n}{n-i} =sum_{i = 0}^{2n}{binom{n}{i}}^2 $$
3
This is a combinatorial proof, which is contrary to what OP wanted.
– Batman
May 13 '14 at 17:43
add a comment |
Note that we can divide the $2n$ objects in $2$ groups including $n$ elements each.
Then to choose $n$ elements out of $2n$ we can choose $0$ objects in the first group and $n$ in the second, or $1$ in the first and $n-1$ in the second and so on, i.e $$binom{2n}{n} = sum_{i = 0}^{2n}binom{n}{i}binom{n}{n-i} =sum_{i = 0}^{2n}{binom{n}{i}}^2 $$
Note that we can divide the $2n$ objects in $2$ groups including $n$ elements each.
Then to choose $n$ elements out of $2n$ we can choose $0$ objects in the first group and $n$ in the second, or $1$ in the first and $n-1$ in the second and so on, i.e $$binom{2n}{n} = sum_{i = 0}^{2n}binom{n}{i}binom{n}{n-i} =sum_{i = 0}^{2n}{binom{n}{i}}^2 $$
answered May 13 '14 at 15:43
WLOG
7,21932258
7,21932258
3
This is a combinatorial proof, which is contrary to what OP wanted.
– Batman
May 13 '14 at 17:43
add a comment |
3
This is a combinatorial proof, which is contrary to what OP wanted.
– Batman
May 13 '14 at 17:43
3
3
This is a combinatorial proof, which is contrary to what OP wanted.
– Batman
May 13 '14 at 17:43
This is a combinatorial proof, which is contrary to what OP wanted.
– Batman
May 13 '14 at 17:43
add a comment |
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