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Number of partitions of n^n into powers of n, also diagonal of A145515 and A196879.
5

%I #17 Feb 05 2017 04:37:10

%S 1,1,4,23,1086,642457,6188114528,1226373476385199,

%T 6071277235712979102634,884267692532264259002637317099,

%U 4362395890943439751990308572939648140812,824887275128335259519795007492785652798382136996397

%N Number of partitions of n^n into powers of n, also diagonal of A145515 and A196879.

%H Alois P. Heinz, <a href="/A145514/b145514.txt">Table of n, a(n) for n = 0..36</a>

%F a(n) = [x^(n^n)] 1/Product_{j>=0}(1-x^(n^j)), n>1.

%e a(2) = 4, because there are 4 partitions of 2^2=4 into powers of 2: [1,1,1,1], [1,1,2], [2,2], [4].

%p g:= proc(b,n,k) option remember; local t; if b<0 then 0 elif b=0 or n=0 or k<=1 then 1 elif b>=n then add(g(b-t, n, k) *binomial(n+1, t) *(-1)^(t+1), t=1..n+1); else g(b-1, n, k) +g(b*k, n-1, k) fi end: a:= n-> g(1,n,n): seq(a(n), n=0..13);

%t g[b_, n_, k_] := g[b, n, k] = Which[b<0, 0, b == 0 || n == 0 || k <= 1, 1, b >= n, Sum[g[b-t, n, k] *Binomial[n+1, t]*(-1)^(t+1), {t, 1, n+1}], True, g[b-1, n, k] + g[b*k, n-1, k]]; a[n_] := g[1, n, n]; Table[a[n], {n, 0, 13}] (* _Jean-François Alcover_, Feb 05 2017, translated from Maple *)

%Y Cf. A145515, A196879, A007318.

%K nonn

%O 0,3

%A _Alois P. Heinz_, Oct 11 2008