A287696 Triangle read by rows, T(n,k) = (n!)^3 * [x^k] [z^n] hypergeom([], [1, 1], z)^x for n>=0, 0<=k<=n.

1, 0, 1, 0, -3, 4, 0, 46, -81, 36, 0, -1899, 3916, -2592, 576, 0, 163476, -375375, 305500, -108000, 14400, 0, -25333590, 63002191, -58725000, 26370000, -5832000, 518400, 0, 6412369860, -16976577828, 17470973569, -9168390000, 2636298000, -400075200, 25401600

Offset: 0

Author

Peter Luschny, May 30 2017

Keywords

• sign
• tabl

Comments

The polynomials Sum_{k=0..n} T(n,k) x^k generate the columns of A287698.

Examples

			0: [1]
1: [0,         1]
2: [0,        -3,        4]
3: [0,        46,      -81,        36]
4: [0,     -1899,     3916,     -2592,      576]
5: [0,    163476,  -375375,    305500,  -108000,    14400]
6: [0, -25333590, 63002191, -58725000, 26370000, -5832000, 518400]
		

Crossrefs

T(n,n) = A001044(n).

Cf. A212856, A212855, A287314, A287698.

Programs

Maple

A287696_row := proc(n) local k; hypergeom([],[1,1],z); series(%^x, z=0, n+1):
n!^3*coeff(%, z, n); seq(coeff(%, x, k), k=0..n) end:
for n from 0 to 8 do A287696_row(n) od;
A287696_poly := proc(n) local k, x; hypergeom([],[1,1],z); series(%^x, z=0, n+1):
unapply(n!^3*coeff(%, z, n), x); end:
for n from 0 to 7 do A287696_poly(n) od;

Mathematica

T[n_, k_] := (n!)^3 SeriesCoefficient[HypergeometricPFQ[{}, {1, 1}, z]^x, {x, 0, k}, {z, 0, n}];
Table[T[n, k], {n, 0, 7}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jun 13 2017 *)

Formula

Sum_{k=0..n} T(n,k) = A000012(n).

Sum_{k=0..n} abs(T(n,k)) = A212856(n) = A212855_row(3).