A293202 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of g.f. Product_{i>0} Sum_{j=0..k} j!*x^(j*i).

1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 3, 2, 0, 1, 1, 3, 2, 2, 0, 1, 1, 3, 8, 6, 3, 0, 1, 1, 3, 8, 6, 7, 4, 0, 1, 1, 3, 8, 30, 13, 12, 5, 0, 1, 1, 3, 8, 30, 13, 24, 13, 6, 0, 1, 1, 3, 8, 30, 133, 48, 37, 22, 8, 0, 1, 1, 3, 8, 30, 133, 48, 61, 46, 26, 10, 0, 1, 1, 3, 8, 30, 133, 768, 181, 142, 98, 42, 12, 0

Offset: 0

Author

Seiichi Manyama, Oct 02 2017

Keywords

• nonn
• tabl

Examples

			Square array begins:
   1, 1, 1,  1,  1, ...
   0, 1, 1,  1,  1, ...
   0, 1, 3,  3,  3, ...
   0, 2, 2,  8,  8, ...
   0, 2, 6,  6, 30, ...
   0, 3, 7, 13, 13, ...
		

Links

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

Crossrefs

Columns k=0..5 give A000007, A000009, A293204, A289485, A289486, A293250.

Rows n=0..1 give A000012, A057427.

Main diagonal gives A161779.

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)-> 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_] := 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 *)