A384362 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where A(n,k) = Sum_{i=0..k*n} 2^i * Sum_{j=0..i} (-1)^j * binomial(i,j) * binomial(i-j,n)^k.

1, 1, 1, 1, 2, 1, 1, 10, 4, 1, 1, 74, 148, 8, 1, 1, 730, 13540, 2440, 16, 1, 1, 9002, 2308756, 3087368, 42256, 32, 1, 1, 133210, 632363044, 10208479240, 778026256, 752800, 64, 1, 1, 2299754, 253970683348, 69754997963528, 52520969994256, 207633589664, 13660480, 128, 1

Offset: 0

Author

Seiichi Manyama, May 27 2025

Keywords

• nonn
• tabl

Examples

			Square array begins:
  1,  1,     1,         1,              1, ...
  1,  2,    10,        74,            730, ...
  1,  4,   148,     13540,        2308756, ...
  1,  8,  2440,   3087368,    10208479240, ...
  1, 16, 42256, 778026256, 52520969994256, ...
		

Crossrefs

Columns k=0..2 give A000012, A000079, A098270.

Rows n=0..1 give A000012, A004123(k+1).

Cf. A262809, A384364.

Programs

PARI

a(n, k) = sum(i=0, k*n, 2^i*sum(j=0, i, (-1)^j*binomial(i, j)*binomial(i-j, n)^k));

Formula

A(n,k) = (1/3) * Sum_{j>=0} (2/3)^j * binomial(j,n)^k.