A293669 Square array A(n,k), n >= 0, k >= 1, read by antidiagonals downwards, where column k is the expansion of e.g.f. exp(Sum_{j=1..k} x^j).

1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 3, 7, 1, 1, 1, 3, 13, 25, 1, 1, 1, 3, 13, 49, 81, 1, 1, 1, 3, 13, 73, 261, 331, 1, 1, 1, 3, 13, 73, 381, 1531, 1303, 1, 1, 1, 3, 13, 73, 501, 2611, 9073, 5937, 1, 1, 1, 3, 13, 73, 501, 3331, 19993, 63393, 26785, 1, 1, 1, 3, 13, 73, 501, 4051, 27553, 165873, 465769, 133651, 1

Offset: 0

Author

Seiichi Manyama, Oct 14 2017

Keywords

• nonn
• tabl

Examples

			Square array begins:
   1,  1,   1,   1,   1, ...
   1,  1,   1,   1,   1, ...
   1,  3,   3,   3,   3, ...
   1,  7,  13,  13,  13, ...
   1, 25,  49,  73,  73, ...
   1, 81, 261, 381, 501, ...
		

Links

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

Crossrefs

Columns k=1..9 give A000012, A047974, A118589, A193930, A193931, A193932, A193933, A306623, A306624.

Rows n=0-1 give A000012.

Main diagonal gives A000262.

Cf. A293718, A293724.

Programs

Maple

A:= proc(n, k) option remember; `if`(n=0, 1, add(
      A(n-j, k)*binomial(n-1, j-1)*j!, j=1..min(n, k)))
    end:
seq(seq(A(n, 1+d-n), n=0..d), d=0..12);  # Alois P. Heinz, Nov 11 2020

Mathematica

A[0, ] = 1; A[n /; n >= 0, k_ /; k >= 1] := A[n, k] = (n-1)!*Sum[j*A[n-j, k]/(n-j)!, {j, 1, Min[k, n]}]; A[, ] = 0;
Table[A[n, d-n+1], {d, 0, 12}, {n, 0, d}] // Flatten (* Jean-François Alcover, Feb 13 2021 *)

Formula

A(0,k) = 1 and A(n,k) = (n-1)! * Sum_{j=1..min(k,n)} j*A(n-j,k)/(n-j)!.