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A229328 Total sum of 6th powers of parts in all partitions of n.

%I A229328 #21 May 28 2018 02:53:02
%S A229328 0,1,66,797,5024,21447,73920,212951,552378,1292410,2838234,5823262,
%T A229328 11464628,21488403,39094986,68600554,117628414,196085189,321067770,
%U A229328 513857202,810429626,1254814258,1918760856,2889290459,4305268546,6331543700,9226796660,13297146272
%N A229328 Total sum of 6th powers of parts in all partitions of n.
%C A229328 The bivariate g.f. for the partition statistic "sum of 6th powers the parts" is G(t,x) = 1/Product_{k>=1}(1 - t^{k^6}*x^k). The g.f. g at the Formula section has been obtained by evaluating dG/dt at t=1. - _Emeric Deutsch_, Dec 06 2015
%H A229328 Alois P. Heinz, <a href="/A229328/b229328.txt">Table of n, a(n) for n = 0..1000</a>
%H A229328 Guo-Niu Han, <a href="https://arxiv.org/abs/0804.1849">An explicit expansion formula for the powers of the Euler Product in terms of partition hook lengths</a>, arXiv:0804.1849 [math.CO], 2008.
%F A229328 a(n) = Sum_{k=1..n} A066633(n,k) * k^6.
%F A229328 G.f.: g(x) = (Sum_{k>=1} k^6*x^k/(1-x^k))/Product_{q>=1} (1-x^q). - _Emeric Deutsch_, Dec 06 2015
%F A229328 a(n) ~ 38880*sqrt(2)*Zeta(7)/Pi^7 * exp(Pi*sqrt(2*n/3)) * n^(5/2). - _Vaclav Kotesovec_, May 28 2018
%p A229328 b:= proc(n, i) option remember; `if`(n=0, [1, 0],
%p A229328       `if`(i<1, [0, 0], `if`(i>n, b(n, i-1),
%p A229328       ((g, h)-> g+h+[0, h[1]*i^6])(b(n, i-1), b(n-i, i)))))
%p A229328     end:
%p A229328 a:= n-> b(n, n)[2]:
%p A229328 seq(a(n), n=0..40);
%p A229328 # second Maple program:
%p A229328 g := (sum(k^6*x^k/(1-x^k), k = 1..100))/(product(1-x^k, k = 1..100)): gser := series(g, x = 0, 45): seq(coeff(gser, x, m), m = 1 .. 40); # _Emeric Deutsch_, Dec 06 2015
%t A229328 (* T = A066633 *) T[n_, n_] = 1; T[n_, k_] /; k < n := T[n, k] = T[n - k, k] + PartitionsP[n - k]; T[_, _] = 0; a[n_] := Sum[T[n, k]*k^6, {k, 1, n}]; Array[a, 40, 0] (* _Jean-François Alcover_, Dec 15 2016 *)
%Y A229328 Column k=6 of A213191.
%K A229328 nonn
%O A229328 0,3
%A A229328 _Alois P. Heinz_, Sep 20 2013