Identities with squares of binomial coefficients

This notebook contains all the calculations from the paper Identities with squares of binomial coefficients (Ars Combinatoria 113 (2014), 377–383), plus a few more. Mathematica 6 or later is needed.

Initialization

The notebook requires the GosperSum package, especially the FactorialSimplify function (abbreviated as FS) implemented there. The package is available from Herbert Wilf’s website http://www.math.upenn.edu/~wilf/progs.html.

Get["http://www.math.upenn.edu/~wilf/gosper.m"];

Basic definitions

This section contains initialization cells that are necessary for all the calculations.

Here are the definitions of the functions (Subscript[S, k])(n) and (Subscript[T, k])(n):

S[0][n_]:=(1/2) (Binomial[4 n,2 n]-((Binomial[2 n,n])^2))

S[1][n_]:=n (Binomial[4 n-1,2 n-1]-3 ((Binomial[2 n-1,n])^2))

T[0][n_]:=(1/2) Binomial[4 n-2,2 n-1]-((Binomial[2 n-1,n-1])^2)

T[1][2]:=0

T[1][n_]:=(1/2) (2 n-1) (Binomial[4 n-3,2 n-2]-2 Binomial[2 n-1,n-1] Binomial[2 n-2,n-2]-((Binomial[2 n-2,n-1])^2))

S[k_][1]:=0

T[k_][2]:=0

For k≥2, the values are calculated using Theorems 1 and 2:

S[k_][n_]:=S[k][n]=4 (n^2) (∑_(i=0)^(k-2))Binomial[k-2,i] T[i][n]

T[k_][n_]:=T[k][n]=(((2 n-1))^2) (∑_(i=0)^(k-2))Binomial[k-2,i] (S[i][n-1]-(((n-2))^i) ((Binomial[2 n-2,n-2])^2))

This is the formula for calculating (Subscript[A, k])(n) from Theorem 4:

A[k_][n_]:=(∑_(j=0)^(k+2))Binomial[k+2,j] (n^(k-j)) (((-1))^j) S[j][n]

The formulas for (Subscript[S, k])(n) and (Subscript[T, k])(n)

S[2][n] //TraditionalForm

T[2][n] //TraditionalForm

S[3][n] //TraditionalForm

T[3][n] //TraditionalForm

S[4][n] //TraditionalForm

T[4][n] //TraditionalForm

These formulas can not be obtained using Gosper's algoritm for indefinite summation:

Table[GosperSum[(l^k) ((Binomial[2 n,l])^2),l],{k,0,10}]

Table[GosperSum[(l^k) ((Binomial[2 n-1,l])^2),l],{k,0,10}]

The formulas for (Subscript[A, k])(n), where k is even

A[0][n] //FS//TraditionalForm

A[2][n]//FS//TraditionalForm

A[4][n]//FS//TraditionalForm

A[6][n]//FS//TraditionalForm

A[8][n]//FS//TraditionalForm

A[10][n]//FS//TraditionalForm

A[12][n]//FS//TraditionalForm

A[14][n]//FS//TraditionalForm

These formulas cannot be obtained using Gosper's algoritm for indefinite summation:

Table[GosperSum[(((l^k) (l^2) ((Binomial[2 n,n-l])^2))/(n^2)),l],{k,0,14,2}]

The formulas for (Subscript[A, k])(n), where k is odd

A[1][n]//FS//TraditionalForm

A[3][n]//FS//TraditionalForm

A[5][n]//FS//TraditionalForm

A[7][n]//FS//TraditionalForm

A[9][n]//FS//TraditionalForm

A[11][n]//FS//TraditionalForm

A[13][n]//FS//TraditionalForm

These formulas can be also obtained using Gosper's algoritm. We get the same results and the computation is much faster:

Table[GosperSum[(((l^k) (l^2) ((Binomial[2 n,n-l])^2))/(n^2)),{l,1,n}],{k,1,15,2}]//FS//TableForm//TraditionalForm

Unfortunately, the algorithm fails for k≥17:

Table[GosperSum[(((l^k) (l^2) ((Binomial[2 n,n-l])^2))/(n^2)),l],{k,17,99,2}]