A304336 T(n, k) = Sum_{j=0..k} (-1)^j*binomial(2*k, j)*(k - j)^(2*n)/(k!)^2, triangle read by rows, n >= 0 and 0 <= k <= n.

1, 0, 1, 0, 1, 3, 0, 1, 15, 10, 0, 1, 63, 140, 35, 0, 1, 255, 1470, 1050, 126, 0, 1, 1023, 14080, 21945, 6930, 462, 0, 1, 4095, 130130, 400400, 252252, 42042, 1716, 0, 1, 16383, 1184820, 6861855, 7747740, 2438436, 240240, 6435

Offset: 0

Author

Peter Luschny, May 11 2018

Keywords

• nonn
• tabl

Examples

			Triangle starts:
[0] 1;
[1] 0, 1;
[2] 0, 1,     3;
[3] 0, 1,    15,      10;
[4] 0, 1,    63,     140,      35;
[5] 0, 1,   255,    1470,    1050,     126;
[6] 0, 1,  1023,   14080,   21945,    6930,     462;
[7] 0, 1,  4095,  130130,  400400,  252252,   42042,   1716;
[8] 0, 1, 16383, 1184820, 6861855, 7747740, 2438436, 240240, 6435;
		

Crossrefs

Row sums are A304338, T(n,n) = A088218 and A001700, T(n,n-1) ~ A002803, T(n,2) ~ A024036, T(n,3) ~ bisection of A174395.

Cf. A304330, A304334.

Programs

Maple

A304336 := (n, k) -> add((-1)^j*binomial(2*k,j)*(k-j)^(2*n), j=0..k)/(k!)^2:
for n from 0 to 8 do seq(A304336(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!)^2;
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!)^2.

T(n, k) = A304334(n, k)/k!.