A293135 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of e.g.f. Product_{i>0} Sum_{j=0..k} x^(j*i)/j!.

1, 1, 0, 1, 1, 0, 1, 1, 2, 0, 1, 1, 3, 12, 0, 1, 1, 3, 12, 48, 0, 1, 1, 3, 13, 72, 360, 0, 1, 1, 3, 13, 72, 480, 2880, 0, 1, 1, 3, 13, 73, 500, 3780, 25200, 0, 1, 1, 3, 13, 73, 500, 4020, 35280, 241920, 0, 1, 1, 3, 13, 73, 501, 4050, 37380, 372960, 2903040, 0

Offset: 0

Author

Seiichi Manyama, Oct 01 2017

Keywords

• nonn
• tabl
• look

Examples

			Square array begins:
   1,   1,   1,   1,   1, ...
   0,   1,   1,   1,   1, ...
   0,   2,   3,   3,   3, ...
   0,  12,  12,  13,  13, ...
   0,  48,  72,  72,  73, ...
   0, 360, 480, 500, 500, ...
		

Links

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

Crossrefs

Columns k=0..5 give A000007, A088311, A293138, A293195, A293196, A293197.

Rows n=0 gives A000012.

Main diagonal gives A000262.

Cf. A293139.

Programs

Maple

b:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0,
      add(b(n-i*j, i-1, k)/j!, j=0..min(k, n/i))))
    end:
A:= (n, k)-> n!*b(n$2, k):
seq(seq(A(n, d-n), n=0..d), d=0..12);  # Alois P. Heinz, Oct 02 2017

Mathematica

b[n_, i_, k_] := b[n, i, k] = If[n == 0, 1, If[i < 1, 0, Sum[b[n - i j, i - 1, k]/j!, {j, 0, Min[k, n/i]}]]];
A[n_, k_] := n! b[n, n, k];
Table[Table[A[n, d - n], {n, 0, d}], {d, 0, 12}] // Flatten (* Jean-François Alcover, Dec 06 2019, after Alois P. Heinz *)