A196879 Square array A(n,k), n>=0, k>=0, read by antidiagonals: A(n,k) is the number of partitions of n^k into powers of k.

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 1, 3, 10, 1, 1, 1, 1, 6, 23, 36, 1, 1, 1, 1, 9, 72, 132, 94, 1, 1, 1, 1, 16, 335, 1086, 729, 284, 1, 1, 1, 1, 36, 2220, 15265, 15076, 3987, 692, 1, 1, 1, 1, 85, 19166, 374160, 642457, 182832, 18687, 1828, 1, 1

Offset

0,13

Links

Alois P. Heinz, Antidiagonals n = 0..44, flattened

Formula

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

Example

A(2,3) = 3, because the number of partitions of 2^3=8 into powers of 3 is 3: [1,1,3,3], [1,1,1,1,1,3], [1,1,1,1,1,1,1,1].
Square array A(n,k) begins:
  1,  1,  1,   1,     1,      1,  ...
  1,  1,  1,   1,     1,      1,  ...
  1,  1,  4,   3,     6,      9,  ...
  1,  1, 10,  23,    72,    335,  ...
  1,  1, 36, 132,  1086,  15265,  ...
  1,  1, 94, 729, 15076, 642457,  ...

Maple

b:= proc(n, j, k) local nn, r;
      if n<0 then 0
    elif j=0 then 1
    elif j=1 then n+1
    elif n<j then b(n, j, k):= b(n-1, j, k) +b(k*n, j-1, k)
             else nn:= 1 +floor(n);
                  r:= n-nn;
                  (nn-j) *binomial(nn, j) *add(binomial(j, h)
                  /(nn-j+h) *b(j-h+r, j, k) *(-1)^h, h=0..j-1)
      fi
    end:
A:= proc(n, k) local s, t;
      if k<2 then return 1 fi;
      s:= floor(n^k/k);
      t:= ilog[k](k*s+1);
      b(s/k^(t-1), t, k)
    end:
seq(seq(A(n, d-n), n=0..d), d=0..15);

Mathematica

a[_, 0] = a[_, 1] = a[0, _] = a[1, _] = 1; a[n_, k_] := SeriesCoefficient[ 1/Product[ (1 - x^(k^j)), {j, 0, n}], {x, 0, n^k}]; Table[a[n - k, k], {n, 0, 10}, {k, n, 0, -1}] // Flatten (* Jean-François Alcover, Dec 09 2013 *)

Crossrefs

Columns k=0+1, 2-10 give: A000012, A196880, A196881, A196882, A196883, A196884, A196885, A196886, A196887, A196888.

Rows n=0+1, 2-10 give: A000012, A196889, A196890, A196891, A196892, A196893, A196894, A196895, A196896, A196897.

Main diagonal gives: A145514.

Cf. A145515.

Sequence in context: A334432 A370130 A010326 * A193349 A053231 A066701

Adjacent sequences: A196876 A196877 A196878 * A196880 A196881 A196882

Keyword

nonn,tabl

Author

Alois P. Heinz, Oct 07 2011

Status

approved