cp's OEIS Frontend

A292194 Sum of n-th powers of products of terms in all partitions of n.

%I A292194 #22 Sep 15 2017 07:37:36
%S A292194 1,1,5,36,610,13225,1173652,92137513,27960729094,14612913824364,
%T A292194 11885159817456154,23676862215173960082,144210774157588042096815,
%U A292194 778807208565930895328294712,15863318347221014170216633451982,908978343753718115412387406378667615
%N A292194 Sum of n-th powers of products of terms in all partitions of n.
%H A292194 Alois P. Heinz, <a href="/A292194/b292194.txt">Table of n, a(n) for n = 0..79</a>
%F A292194 a(n) = [x^n] Product_{k=1..n} 1/(1 - k^n*x^k).
%F A292194 From _Vaclav Kotesovec_, Sep 15 2017: (Start)
%F A292194 a(n) ~ 3^(n^2/3)               if mod(n,3)=0
%F A292194 a(n) ~ 3^(n*(n-4)/3)*2^(2*n+1) if mod(n,3)=1
%F A292194 a(n) ~ 3^(n*(n-2)/3)*2^n       if mod(n,3)=2
%F A292194 (End)
%e A292194 5 = 4 + 1 = 3 + 2 = 3 + 1 + 1 = 2 + 2 + 1 = 2 + 1 + 1 + 1 = 1 + 1 + 1 + 1 + 1.
%e A292194 So a(5) = 5^5 + (4*1)^5 + (3*2)^5 + (3*1*1)^5 + (2*2*1)^5 + (2*1*1*1)^5 + (1*1*1*1*1)^5 = 13225.
%p A292194 b:= proc(n, i, k) option remember; `if`(n=0 or i=1, 1,
%p A292194       `if`(i>n, 0, i^k*b(n-i, i, k))+b(n, i-1, k))
%p A292194     end:
%p A292194 a:= n-> b(n$3):
%p A292194 seq(a(n), n=0..20);  # _Alois P. Heinz_, Sep 11 2017
%t A292194 nmax = 20; Table[SeriesCoefficient[Product[1/(1 - k^n*x^k), {k, 1, n}], {x, 0, n}], {n, 0, nmax}] (* _Vaclav Kotesovec_, Sep 15 2017 *)
%o A292194 (PARI) {a(n) = polcoeff(1/prod(k=1, n, 1-k^n*x^k+x*O(x^n)), n)}
%Y A292194 Main diagonal of A292193.
%Y A292194 Cf. A292190.
%K A292194 nonn
%O A292194 0,3
%A A292194 _Seiichi Manyama_, Sep 11 2017