A361600 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,k*j)/j!.

1, 1, 2, 1, 2, 5, 1, 2, 7, 16, 1, 2, 9, 34, 65, 1, 2, 11, 58, 209, 326, 1, 2, 13, 88, 473, 1546, 1957, 1, 2, 15, 124, 881, 4626, 13327, 13700, 1, 2, 17, 166, 1457, 10526, 52537, 130922, 109601, 1, 2, 19, 214, 2225, 20326, 145867, 677594, 1441729, 986410

Offset

0,3

Links

Seiichi Manyama, Antidiagonals n = 0..139, flattened

Formula

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

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

Example

Square array begins:
    1,    1,    1,     1,     1,     1, ...
    2,    2,    2,     2,     2,     2, ...
    5,    7,    9,    11,    13,    15, ...
   16,   34,   58,    88,   124,   166, ...
   65,  209,  473,   881,  1457,  2225, ...
  326, 1546, 4626, 10526, 20326, 35226, ...

Prog

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

Crossrefs

Columns k=0..3 give A000522, A002720, A361598, A361599.

Main diagonal gives A361607.

Cf. A293012.

Sequence in context: A334165 A370887 A210223 * A242598 A225568 A291694

Adjacent sequences: A361597 A361598 A361599 * A361601 A361602 A361603

Keyword

nonn,tabl

Author

Seiichi Manyama, Mar 17 2023

Status

approved