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A292190 Sum of n-th powers of products of terms in all partitions of n into distinct parts.

%I A292190 #22 Sep 14 2017 08:17:10
%S A292190 1,1,4,35,337,11925,371081,49032439,3545396034,3416952655320,
%T A292190 749189363202730,598250899004413536,2383502427069445040595,
%U A292190 1729793152213690218766715,131751643363739706679145099315,271212858254426215135033141804302
%N A292190 Sum of n-th powers of products of terms in all partitions of n into distinct parts.
%H A292190 Alois P. Heinz, <a href="/A292190/b292190.txt">Table of n, a(n) for n = 0..98</a>
%F A292190 a(n) = [x^n] Product_{k=1..n} (1 + k^n*x^k).
%e A292190 5 = 4 + 1 = 3 + 2. So a(5) = 5^5 + (4*1)^5 + (3*2)^5 = 11925.
%p A292190 b:= proc(n, i, k) option remember; (m->
%p A292190       `if`(m<n, 0, `if`(n=m, i!^k, b(n, i-1, k)+
%p A292190       `if`(i>n, 0, i^k*b(n-i, i-1, k)))))(i*(i+1)/2)
%p A292190     end:
%p A292190 a:= n-> b(n$3):
%p A292190 seq(a(n), n=0..20);  # _Alois P. Heinz_, Sep 11 2017
%t A292190 nmax = 15; Table[SeriesCoefficient[Product[(1 + k^n*x^k), {k, 1, nmax}], {x, 0, n}], {n, 0, nmax}] (* _Vaclav Kotesovec_, Sep 12 2017 *)
%o A292190 (PARI) {a(n) = polcoeff(prod(k=1, n, 1+k^n*x^k+x*O(x^n)), n)}
%Y A292190 Main diagonal of A292189.
%Y A292190 Cf. A265949, A292167, A292194, A292305, A292306.
%K A292190 nonn
%O A292190 0,3
%A A292190 _Seiichi Manyama_, Sep 11 2017