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A229325 Total sum of cubes of parts in all partitions of n.

%I A229325 #28 May 28 2018 02:49:10
%S A229325 0,1,10,39,122,287,660,1281,2486,4392,7686,12628,20790,32471,50694,
%T A229325 76560,115038,168333,245784,350896,499620,699468,975150,1341077,
%U A229325 1838550,2490092,3361260,4494084,5986750,7909231,10416300,13616768,17745948,22983345,29672974
%N A229325 Total sum of cubes of parts in all partitions of n.
%C A229325 The bivariate g.f. for the partition statistic "sum of cubes of the parts" is G(t,x) = 1/Product_{k>=1}(1 - t^{k^3}*x^k). The g.f. g given in the Formula section was obtained by evaluating dG/dt at t=1. - _Emeric Deutsch_, Dec 06 2015
%H A229325 Alois P. Heinz, <a href="/A229325/b229325.txt">Table of n, a(n) for n = 0..8500</a>
%H A229325 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 A229325 a(n) = Sum_{k=1..n} A066633(n,k) * k^3.
%F A229325 G.f.: g(x) = (Sum_{k>=1} k^3*x^k/(1-x^k))/Product_{q>=1} (1-x^q). - _Emeric Deutsch_, Dec 06 2015
%F A229325 a(n) ~ sqrt(3)/5 * exp(Pi*sqrt(2*n/3)) * n. - _Vaclav Kotesovec_, May 28 2018
%p A229325 b:= proc(n, i) option remember; `if`(n=0, [1, 0],
%p A229325       `if`(i<1, [0, 0], `if`(i>n, b(n, i-1),
%p A229325       ((g, h)-> g+h+[0, h[1]*i^3])(b(n, i-1), b(n-i, i)))))
%p A229325     end:
%p A229325 a:= n-> b(n, n)[2]:
%p A229325 seq(a(n), n=0..40);
%t A229325 Table[Total[Flatten[IntegerPartitions[n]^3]],{n,0,40}] (* _Harvey P. Dale_, May 01 2016 *)
%t A229325 b[n_, i_] := b[n, i] = If[n==0, {1, 0}, If[i<1, {0, 0}, If[i>n, b[n, i-1], Function[{g, h}, g + h + {0, h[[1]]*i^3}][b[n, i-1], b[n-i, i]]]]];
%t A229325 a[n_] := b[n, n][[2]];
%t A229325 Table[a[n], {n, 0, 40}] (* _Jean-François Alcover_, Aug 30 2016, after _Alois P. Heinz_ *)
%Y A229325 Column k=3 of A213191.
%K A229325 nonn
%O A229325 0,3
%A A229325 _Alois P. Heinz_, Sep 20 2013