A361277 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,k) = n! * Sum_{j=0..n} binomial(k*j,n-j)/j!.

1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 5, 7, 1, 1, 1, 7, 19, 25, 1, 1, 1, 9, 37, 97, 81, 1, 1, 1, 11, 61, 241, 581, 331, 1, 1, 1, 13, 91, 481, 1981, 3661, 1303, 1, 1, 1, 15, 127, 841, 4881, 17551, 26335, 5937, 1, 1, 1, 17, 169, 1345, 10001, 55321, 171697, 202049, 26785, 1

Offset: 0

Author

Seiichi Manyama, Mar 06 2023

Keywords

• nonn
• tabl

Examples

			Square array begins:
  1,  1,   1,    1,    1,     1, ...
  1,  1,   1,    1,    1,     1, ...
  1,  3,   5,    7,    9,    11, ...
  1,  7,  19,   37,   61,    91, ...
  1, 25,  97,  241,  481,   841, ...
  1, 81, 581, 1981, 4881, 10001, ...
		

Crossrefs

Columns k=0..4 give A000012, A047974, A361278, A361279, A361280.

Main diagonal gives A361281.

Cf. A293012.

Programs

PARI

T(n, k) = n!*sum(j=0, n, binomial(k*j, n-j)/j!);

Formula

E.g.f. of column k: exp(x * (1+x)^k).

T(0,k) = 1; T(n,k) = (n-1)! * Sum_{j=1..n} j * binomial(k,j-1) * T(n-j,k)/(n-j)!.