A381512 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where A(n,k) = (2*n+k)!/k! * [x^(2*n+k)] sinh(x)^k.

1, 1, 0, 1, 1, 0, 1, 4, 1, 0, 1, 10, 16, 1, 0, 1, 20, 91, 64, 1, 0, 1, 35, 336, 820, 256, 1, 0, 1, 56, 966, 5440, 7381, 1024, 1, 0, 1, 84, 2352, 24970, 87296, 66430, 4096, 1, 0, 1, 120, 5082, 90112, 631631, 1397760, 597871, 16384, 1, 0, 1, 165, 10032, 273988, 3331328, 15857205, 22368256, 5380840, 65536, 1, 0

Offset: 0

Author

Seiichi Manyama, May 11 2025

Keywords

• nonn
• tabl

Examples

			Square array begins:
  1, 1,    1,     1,       1,        1, ...
  0, 1,    4,    10,      20,       35, ...
  0, 1,   16,    91,     336,      966, ...
  0, 1,   64,   820,    5440,    24970, ...
  0, 1,  256,  7381,   87296,   631631, ...
  0, 1, 1024, 66430, 1397760, 15857205, ...
		

Crossrefs

Columns k=0..7 give A000007, A000012, A000302, A002452(n+1), A166984, A002453, 4^n * A002451(n), A381513.

Main diagonal gives A383837.

Cf. A136630, A160562, A286899.

Programs

PARI

a(n, k) = (2*n+k)!/k!*polcoef(sinh(x+x*O(x^(2*n+k)))^k, 2*n+k);

Formula

G.f. of column k: 1/Product_{j=0..floor(k/2)} (1 - (k-2*j)^2*x).

A(n,k) = k^2 * A(n-1,k) + A(n,k-2) for k > 1.

A(n,k) = (1/(2^k*k!)) * Sum_{j=0..k} (-1)^j * (k-2*j)^(2*n+k) * binomial(k,j).