A384364 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where A(n,k) = Sum_{i=0..k*n} 3^i * Sum_{j=0..i} (-1)^j * binomial(i,j) * binomial(i-j,n)^k.

1, 1, 1, 1, 3, 1, 1, 21, 9, 1, 1, 219, 657, 27, 1, 1, 3045, 119241, 22869, 81, 1, 1, 52923, 40365873, 80850987, 836001, 243, 1, 1, 1103781, 21955523049, 747786838869, 60579666801, 31436181, 729, 1, 1, 26857659, 17512689629457, 14298291269335467, 16117269494868801, 48066954848379, 1204022961, 2187, 1

Offset: 0

Author

Seiichi Manyama, May 27 2025

Keywords

• nonn
• tabl

Examples

			Square array begins:
  1,  1,      1,           1,                 1, ...
  1,  3,     21,         219,              3045, ...
  1,  9,    657,      119241,          40365873, ...
  1, 27,  22869,    80850987,      747786838869, ...
  1, 81, 836001, 60579666801, 16117269494868801, ...
		

Crossrefs

Columns k=0..2 give A000012, A000244, 3^n * A084768(n).

Rows n=0..1 give A000012, A032033.

Cf. A262809, A384362.

Programs

PARI

a(n, k) = sum(i=0, k*n, 3^i*sum(j=0, i, (-1)^j*binomial(i, j)*binomial(i-j, n)^k));

Formula

A(n,k) = (1/4) * Sum_{j>=0} (3/4)^j * binomial(j,n)^k.