A304334 T(n, k) = Sum_{j=0..k} (-1)^j*binomial(2*k, j)*(k - j)^(2*n)/k!, triangle read by rows, n >= 0 and 0 <= k <= n.

1, 0, 1, 0, 1, 6, 0, 1, 30, 60, 0, 1, 126, 840, 840, 0, 1, 510, 8820, 25200, 15120, 0, 1, 2046, 84480, 526680, 831600, 332640, 0, 1, 8190, 780780, 9609600, 30270240, 30270240, 8648640, 0, 1, 32766, 7108920, 164684520, 929728800, 1755673920, 1210809600, 259459200

Offset: 0

Author

Peter Luschny, May 11 2018

Keywords

• nonn
• tabl

Examples

			Triangle starts:
[0] 1
[1] 0, 1
[2] 0, 1,     6
[3] 0, 1,    30,      60
[4] 0, 1,   126,     840,       840
[5] 0, 1,   510,    8820,     25200,     15120
[6] 0, 1,  2046,   84480,    526680,    831600,     332640
[7] 0, 1,  8190,  780780,   9609600,  30270240,   30270240,    8648640
[8] 0, 1, 32766, 7108920, 164684520, 929728800, 1755673920, 1210809600, 259459200
		

Crossrefs

Row sums are bisection of A081562, T(n,n) ~ A000407, T(n,n-1) ~ A048854(n,2), T(n,2) ~ A002446.

Cf. A304330, A304336.

Programs

Maple

A304334 := (n, k) -> add((-1)^j*binomial(2*k,j)*(k-j)^(2*n), j=0..k)/k!:
for n from 0 to 8 do seq(A304334(n, k), k=0..n) od;

PARI

T(n, k) = sum(j=0, k, (-1)^j*binomial(2*k, j)*(k - j)^(2*n))/k!;
tabl(nn) = for (n=0, nn, for (k=0, n, print1(T(n,k), ", ")); print); \\ Michel Marcus, May 11 2018

Formula

T(n, k) = A304330(n, k) / k!.