A259799 Array read by antidiagonals upwards: T(n,k) = number of partitions of k^n into n-th powers (n>=1, k>=0).

1, 1, 1, 1, 1, 2, 1, 1, 2, 3, 1, 1, 2, 4, 5, 1, 1, 2, 5, 8, 7, 1, 1, 2, 7, 17, 19, 11, 1, 1, 2, 9, 36, 62, 43, 15, 1, 1, 2, 13, 88, 253, 258, 98, 22, 1, 1, 2, 19, 218, 1104, 1886, 1050, 220, 30, 1, 1, 2, 27, 550, 5082, 15772, 14800, 4365, 504, 42, 1, 1, 2, 40, 1413, 24119, 140549, 241582, 118238, 18012, 1116, 56

Offset: 1

Author

N. J. A. Sloane, Jul 06 2015

Keywords

• nonn
• tabl

Examples

			The array begins:
  1, 1, 2, 3, 5, 7, 11, 15, 22, 30, ...
  1, 1, 2, 4, 8, 19, 43, 98, 220, 504, ...
  1, 1, 2, 5, 17, 62, 258, 1050, 4365, 18012, ...
  1, 1, 2, 7, 36, 253, 1886, 14800, 118238, ...
  1, 1, 2, 9, 88, 1104, 15772, 241582, ...
  ...
		

Links

• Alois P. Heinz, Antidiagonals n = 1..16, flattened
• H. L. Fisher, Letter to N. J. A. Sloane, Mar 16 1989

Crossrefs

Rows: A000041, A037444, A259792-A259795.

Columns: A259796, A027601, A259797, A259798.

T(n,n) gives A331402.

Programs

Maple

b:= proc(n, i, k) option remember; `if`(n=0 or i=1, 1,
      `if`(i=2, 1+iquo(n, i^k), b(n, i-1, k)+
      `if`(i^k>n, 0, b(n-i^k, i, k))))
    end:
T:= (n, k)-> b(k^n, k, n):
seq(seq(T(d-k, k), k=0..d-1), d=1..12);  # Alois P. Heinz, Jul 10 2015

Mathematica

b[n_, i_, k_] := b[n, i, k] = If[n==0 || i==1, 1, If[i==2, 1+Quotient[n, i^k], b[n, i-1, k] + If[i^k>n, 0, b[n-i^k, i, k]]]]; T[n_, k_] := b[k^n, k, n]; Table[ Table[ T[d-k, k], {k, 0, d-1}], {d, 1, 12}] // Flatten (* Jean-François Alcover, Jul 15 2015, after Alois P. Heinz *)

Extensions

More terms from Alois P. Heinz, Jul 10 2015