A361616 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,k) = n! * Sum_{j=0..n} binomial(n+(k-1)*(j+1),n-j)/j!.

1, 1, 1, 1, 2, 1, 1, 3, 7, 1, 1, 4, 15, 34, 1, 1, 5, 25, 103, 209, 1, 1, 6, 37, 214, 885, 1546, 1, 1, 7, 51, 373, 2293, 9051, 13327, 1, 1, 8, 67, 586, 4721, 29176, 106843, 130922, 1, 1, 9, 85, 859, 8481, 70981, 427189, 1425495, 1441729, 1

Offset: 0

Author

Seiichi Manyama, Mar 18 2023

Keywords

• nonn
• tabl

Examples

			Square array begins:
  1,    1,    1,     1,     1,      1, ...
  1,    2,    3,     4,     5,      6, ...
  1,    7,   15,    25,    37,     51, ...
  1,   34,  103,   214,   373,    586, ...
  1,  209,  885,  2293,  4721,   8481, ...
  1, 1546 ,9051, 29176, 70981, 146046, ...
		

Crossrefs

Columns k=0..3 give A000012, A002720, A343884, A351767.

Main diagonal gives A361617.

Cf. A293985, A361600.

Programs

PARI

T(n,k) = n! * sum(j=0, n, binomial(n+(k-1)*(j+1), n-j)/j!);

Formula

E.g.f. of column k: exp( x/(1-x)^k ) / (1-x)^k.

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