A292189 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of Product_{j>=1} (1 + j^k*x^j).

1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 4, 5, 2, 1, 1, 8, 13, 7, 3, 1, 1, 16, 35, 25, 15, 4, 1, 1, 32, 97, 91, 77, 25, 5, 1, 1, 64, 275, 337, 405, 161, 43, 6, 1, 1, 128, 793, 1267, 2177, 1069, 393, 64, 8, 1, 1, 256, 2315, 4825, 11925, 7313, 3799, 726, 120, 10

Offset

0,9

Links

Alois P. Heinz, Rows n = 0..150, flattened

Example

Square array begins:
   1, 1,  1,  1,   1, ...
   1, 1,  1,  1,   1, ...
   1, 2,  4,  8,  16, ...
   2, 5, 13, 35,  97, ...
   2, 7, 25, 91, 337, ...

Maple

b:= proc(n, i, k) option remember; (m->
      `if`(m<n, 0, `if`(n=m, i!^k, b(n, i-1, k)+
      `if`(i>n, 0, i^k*b(n-i, i-1, k)))))(i*(i+1)/2)
    end:
A:= (n, k)-> b(n$2, k):
seq(seq(A(n, d-n), n=0..d), d=0..14);  # Alois P. Heinz, Sep 11 2017

Mathematica

m = 14;
col[k_] := col[k] = Product[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 *)

Crossrefs

Columns k=0..5 give A000009, A022629, A092484, A265840, A265841, A265842.

Rows 0+1, 2, 3 give A000012, A000079, A007689.

Main diagonal gives A292190.

Cf. A292166.

Sequence in context: A301895 A229054 A133135 * A284992 A191687 A322190

Adjacent sequences: A292186 A292187 A292188 * A292190 A292191 A292192

Keyword

nonn,tabl

Author

Seiichi Manyama, Sep 11 2017

Status

approved