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A229329 Total sum of 7th powers of parts in all partitions of n.

%I A229329 #21 May 28 2018 02:53:59
%S A229329 0,1,130,2319,18962,99407,400620,1323441,3835406,9924912,23736846,
%T A229329 52729348,111173790,222415631,428578374,794363760,1430855958,
%U A229329 2500747293,4274146464,7130355736,11681752260,18764913468,29690460150,46211761397,71016916110,107622522692
%N A229329 Total sum of 7th powers of parts in all partitions of n.
%C A229329 The bivariate g.f. for the partition statistic "sum of 7th powers of the parts" is G(t,x) = 1/Product_{k>=1}(1 - t^{k^7}*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 A229329 Alois P. Heinz, <a href="/A229329/b229329.txt">Table of n, a(n) for n = 0..1000</a>
%H A229329 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 A229329 a(n) = Sum_{k=1..n} A066633(n,k) * k^7.
%F A229329 G.f.: g(x) = (Sum_{k>=1} k^7*x^k/(1-x^k))/Product_{q>=1}(1-x^q). - _Emeric Deutsch_, Dec 06 2015
%F A229329 a(n) ~ 288*sqrt(3)/5 * exp(Pi*sqrt(2*n/3)) * n^3. - _Vaclav Kotesovec_, May 28 2018
%p A229329 b:= proc(n, i) option remember; `if`(n=0, [1, 0],
%p A229329       `if`(i<1, [0, 0], `if`(i>n, b(n, i-1),
%p A229329       ((g, h)-> g+h+[0, h[1]*i^7])(b(n, i-1), b(n-i, i)))))
%p A229329     end:
%p A229329 a:= n-> b(n, n)[2]:
%p A229329 seq(a(n), n=0..40);
%p A229329 # second Maple program:
%p A229329 g := (sum(k^7*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 A229329 (* 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^7, {k, 1, n}]; Array[a, 40, 0] (* _Jean-François Alcover_, Dec 15 2016 *)
%Y A229329 Column k=7 of A213191.
%K A229329 nonn
%O A229329 0,3
%A A229329 _Alois P. Heinz_, Sep 20 2013