A145515 Square array A(n,k), n>=0, k>=0, read by antidiagonals: A(n,k) is the number of partitions of k^n into powers of k.

1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 4, 1, 1, 1, 2, 5, 10, 1, 1, 1, 2, 6, 23, 36, 1, 1, 1, 2, 7, 46, 239, 202, 1, 1, 1, 2, 8, 82, 1086, 5828, 1828, 1, 1, 1, 2, 9, 134, 3707, 79326, 342383, 27338, 1, 1, 1, 2, 10, 205, 10340, 642457, 18583582, 50110484, 692004, 1, 1, 1, 2, 11, 298, 24901, 3649346, 446020582, 14481808030, 18757984046, 30251722, 1, 1

Offset

0,8

Links

Alois P. Heinz, Antidiagonals n = 0..40, flattened

Formula

See program.

For k>1: A(n,k) = [x^(k^n)] 1/Product_{j>=0} (1-x^(k^j)).

Example

A(2,3) = 5, because there are 5 partitions of 3^2=9 into powers of 3: [1,1,1,1,1,1,1,1,1], [1,1,1,1,1,1,3], [1,1,1,3,3], [3,3,3], [9].
Square array A(n,k) begins:
  1,  1,   1,    1,     1,      1,  ...
  1,  1,   2,    2,     2,      2,  ...
  1,  1,   4,    5,     6,      7,  ...
  1,  1,  10,   23,    46,     82,  ...
  1,  1,  36,  239,  1086,   3707,  ...
  1,  1, 202, 5828, 79326, 642457,  ...

Maple

b:= proc(n, j, k) local nn;
      nn:= n+1;
      if n<0  then 0
    elif j=0  or n=0 or k<=1 then 1
    elif j=1  then nn
    elif n>=j then (nn-j) *binomial(nn, j) *add(binomial(j, h)
                   /(nn-j+h) *b(j-h-1, j, k) *(-1)^h, h=0..j-1)
              else b(n, j, k):= b(n-1, j, k) +b(k*n, j-1, k)
      fi
    end:
A:= (n, k)-> b(1, n, k):
seq(seq(A(n, d-n), n=0..d), d=0..13);

Mathematica

b[n_, j_, k_] := Module[{nn = n+1}, Which[n < 0, 0, j == 0 || n == 0 || k <= 1, 1, j == 1, nn, n >= j, (nn-j)*Binomial[nn, j]*Sum[Binomial[j, h]/(nn-j+h)* b[j-h-1, j, k]*(-1)^h, {h, 0, j-1}], True, b[n, j, k] = b[n-1, j, k] + b[k*n, j-1, k] ] ]; a[n_, k_] := b[1, n, k]; Table[Table[a[n, d-n], {n, 0, d}], {d, 0, 13}] // Flatten (* Jean-François Alcover, Dec 12 2013, translated from Maple *)

Crossrefs

Columns k=0+1, 2-10 give: A000012, A002577, A078125, A078537, A111822, A111827, A111832, A111837, A145512, A145513.

Row n=3 gives: A189890(k+1).

Main diagonal gives: A145514.

Cf. A007318.

Sequence in context: A295679 A287214 A287216 * A267383 A332648 A272896

Adjacent sequences: A145512 A145513 A145514 * A145516 A145517 A145518

Keyword

nonn,tabl

Author

Alois P. Heinz, Oct 11 2008

Extensions

Edited by Alois P. Heinz, Jan 12 2011

Status

approved