A292193 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of Product_{j>=1} 1/(1 - j^k*x^j).

1, 1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 5, 6, 5, 1, 1, 9, 14, 14, 7, 1, 1, 17, 36, 46, 25, 11, 1, 1, 33, 98, 164, 107, 56, 15, 1, 1, 65, 276, 610, 505, 352, 97, 22, 1, 1, 129, 794, 2324, 2531, 2474, 789, 198, 30, 1, 1, 257, 2316, 8986, 13225, 18580, 7273, 2314, 354, 42

Offset: 0

Author

Seiichi Manyama, Sep 11 2017

Keywords

• nonn
• tabl

Examples

			Square array begins:
   1,  1,  1,   1,   1, ...
   1,  1,  1,   1,   1, ...
   2,  3,  5,   9,  17, ...
   3,  6, 14,  36,  98, ...
   5, 14, 46, 164, 610, ...
		

Links

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

Crossrefs

Columns k=0..5 give A000041, A006906, A077335, A265837, A265838, A265839.

Rows 0+1, 2 give A000012, A000051.

Main diagonal gives A292194.

Cf. A292166.

Programs

Maple

b:= proc(n, i, k) option remember; `if`(n=0 or i=1, 1,
      `if`(i>n, 0, i^k*b(n-i, i, k))+b(n, i-1, k))
    end:
A:= (n, k)-> b(n$2, k):
seq(seq(A(n, d-n), n=0..d), d=0..12);  # Alois P. Heinz, Sep 11 2017

Mathematica

m = 12;
col[k_] := col[k] = Product[1/(1 - j^k*x^j), {j, 1, m}] + O[x]^(m+1) // CoefficientList[#, x]&;
A[n_, k_] := col[k][[n+1]];
Table[A[n, d-n], {d, 0, m}, {n, 0, d}] // Flatten (* Jean-François Alcover, Feb 11 2021 *)

Formula

A(0,k) = 1 and A(n,k) = (1/n) * Sum_{j=1..n} (Sum_{d|j} d^(1+k*j/d)) * A(n-j,k) for n > 0. - Seiichi Manyama, Nov 02 2017