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
Examples
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, ...
Links
- Alois P. Heinz, Antidiagonals n = 0..44, flattened
Crossrefs
Programs
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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 -
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 *)
Formula
For k>1: A(n,k) = [x^(n^k)] 1/Product_{j>=0}(1-x^(k^j)).