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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.

%I A145515 #30 Mar 19 2019 12:47:42
%S A145515 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,
%T A145515 239,202,1,1,1,2,8,82,1086,5828,1828,1,1,1,2,9,134,3707,79326,342383,
%U A145515 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
%N 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.
%H A145515 Alois P. Heinz, <a href="/A145515/b145515.txt">Antidiagonals n = 0..40, flattened</a>
%F A145515 See program.
%F A145515 For k>1: A(n,k) = [x^(k^n)] 1/Product_{j>=0} (1-x^(k^j)).
%e A145515 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].
%e A145515 Square array A(n,k) begins:
%e A145515   1,  1,   1,    1,     1,      1,  ...
%e A145515   1,  1,   2,    2,     2,      2,  ...
%e A145515   1,  1,   4,    5,     6,      7,  ...
%e A145515   1,  1,  10,   23,    46,     82,  ...
%e A145515   1,  1,  36,  239,  1086,   3707,  ...
%e A145515   1,  1, 202, 5828, 79326, 642457,  ...
%p A145515 b:= proc(n, j, k) local nn;
%p A145515       nn:= n+1;
%p A145515       if n<0  then 0
%p A145515     elif j=0  or n=0 or k<=1 then 1
%p A145515     elif j=1  then nn
%p A145515     elif n>=j then (nn-j) *binomial(nn, j) *add(binomial(j, h)
%p A145515                    /(nn-j+h) *b(j-h-1, j, k) *(-1)^h, h=0..j-1)
%p A145515               else b(n, j, k):= b(n-1, j, k) +b(k*n, j-1, k)
%p A145515       fi
%p A145515     end:
%p A145515 A:= (n, k)-> b(1, n, k):
%p A145515 seq(seq(A(n, d-n), n=0..d), d=0..13);
%t A145515 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 *)
%Y A145515 Columns k=0+1, 2-10 give: A000012, A002577, A078125, A078537, A111822, A111827, A111832, A111837, A145512, A145513.
%Y A145515 Row n=3 gives: A189890(k+1).
%Y A145515 Main diagonal gives: A145514.
%Y A145515 Cf. A007318.
%K A145515 nonn,tabl
%O A145515 0,8
%A A145515 _Alois P. Heinz_, Oct 11 2008
%E A145515 Edited by _Alois P. Heinz_, Jan 12 2011