A304330 T(n, k) = Sum_{j=0..k} (-1)^j*binomial(2*k, j)*(k - j)^(2*n), triangle read by rows, n >= 0 and 0 <= k <= n.

1, 0, 1, 0, 1, 12, 0, 1, 60, 360, 0, 1, 252, 5040, 20160, 0, 1, 1020, 52920, 604800, 1814400, 0, 1, 4092, 506880, 12640320, 99792000, 239500800, 0, 1, 16380, 4684680, 230630400, 3632428800, 21794572800, 43589145600, 0, 1, 65532, 42653520, 3952428480, 111567456000, 1264085222400, 6102480384000, 10461394944000

Offset: 0

Author

Peter Luschny, May 11 2018

Keywords

• nonn
• tabl

Examples

			Triangle starts:
  [0] 1;
  [1] 0, 1;
  [2] 0, 1,    12;
  [3] 0, 1,    60,     360;
  [4] 0, 1,   252,    5040,     20160;
  [5] 0, 1,  1020,   52920,    604800,    1814400;
  [6] 0, 1,  4092,  506880,  12640320,   99792000,   239500800;
  [7] 0, 1, 16380, 4684680, 230630400, 3632428800, 21794572800, 43589145600;
		

Links

• José L. Cereceda, Sums of powers of integers and the sequence A304330, arXiv:2405.05268 [math.GM], 2024.

Crossrefs

Row sums are A100872, T(n,2) = A058896, T(n,n) = A002674, T(n,n-1)= A091032.

Cf. A304334, A304336.

Programs

Maple

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

PARI

T(n, k) = sum(j=0, k, (-1)^j*binomial(2*k, j)*(k - j)^(2*n)); \\ Michel Marcus, Aug 03 2025