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A252761 Total sum of n-th powers of parts in all partitions of n.

%I A252761 #12 Jul 21 2018 07:23:38
%S A252761 0,1,6,39,392,4775,73920,1323441,27530298,644749920,16877063274,
%T A252761 486936848068,15373069624220,526779275391863,19477946814752586,
%U A252761 772859962684631760,32758854443379036238,1477205045259973740909,70613293111837146235770,3566735926987461858837256
%N A252761 Total sum of n-th powers of parts in all partitions of n.
%H A252761 Alois P. Heinz, <a href="/A252761/b252761.txt">Table of n, a(n) for n = 0..200</a>
%F A252761 a(n) = Sum_{j=1..n} A066633(n,j) * j^n.
%F A252761 a(n) ~ c * n^n, where c = 1/QPochhammer(exp(-1)) = 1.98244090741287370368568246556131201568288277843252568635840026086375046496... - _Vaclav Kotesovec_, May 28 2018, updated Jul 21 2018
%p A252761 b:= proc(n, p, k) option remember; `if`(n=0, [1, 0], `if`(p<1, [0, 0],
%p A252761       add((l->`if`(m=0, l, l+[0, l[1]*p^k*m]))
%p A252761           (b(n-p*m, p-1, k)), m=0..n/p)))
%p A252761     end:
%p A252761 a:= n-> b(n$3)[2]:
%p A252761 seq(a(n), n=0..20);
%t A252761 b[n_, p_, k_] := b[n, p, k] = If[n == 0, {1, 0}, If[p < 1, {0, 0}, Sum[ Function[l, If[m == 0, l, l + {0, l[[1]]*p^k*m}]][b[n - p*m, p - 1, k]], {m, 0, n/p}]]]; a[n_] := b[n, n, n][[2]]; Table[a[n], {n, 0, 20}] (* _Jean-François Alcover_, Feb 07 2017, translated from Maple *)
%Y A252761 Main diagonal of A213191.
%Y A252761 Cf. A066633.
%K A252761 nonn
%O A252761 0,3
%A A252761 _Alois P. Heinz_, Dec 21 2014