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A264392 Number of perfect cube parts in all partitions of n.

%I A264392 #17 Nov 14 2020 06:25:49
%S A264392 0,1,2,4,7,12,19,30,46,68,99,142,200,279,384,523,707,946,1256,1656,
%T A264392 2169,2822,3652,4699,6017,7666,9725,12282,15452,19362,24176,30080,
%U A264392 37307,46117,56843,69854,85613,104640,127578,155150,188249,227872,275242,331738,399027,478988
%N A264392 Number of perfect cube parts in all partitions of n.
%C A264392 a(n) = Sum_{k=0..n} k*A264391(n,k).
%H A264392 Alois P. Heinz, <a href="/A264392/b264392.txt">Table of n, a(n) for n = 0..1000</a>
%F A264392 G.f.: ( Sum_{i>0} x^(h(i))/(1-x^(h(i))) ) / ( Product_{i>0} 1-x^i ), where h(i) = i^3.
%e A264392 a(4) = 7 because the partitions of 4 are [4],[3,1'],[2,2],[2,1',1'], and  [1',1',1',1'], where the perfect cube parts are marked.
%p A264392 h := proc (i) options operator, arrow: i^3 end proc: g := (sum(x^h(i)/(1-x^h(i)), i = 1 .. 100))/(product(1-x^i, i = 1 .. 100)): hser := series(g, x = 0, 55): seq(coeff(hser, x, n), n = 0 .. 50);
%t A264392 cnt[P_List] := Count[P, p_ /; IntegerQ[p^(1/3)]];
%t A264392 a[n_] := a[n] = cnt /@ IntegerPartitions[n] // Total;
%t A264392 Table[Print[n, " ", a[n]]; a[n], {n, 0, 50}];
%t A264392 (* or: *)
%t A264392 m = 50;
%t A264392 CoefficientList[Sum[x^(i^3)/(1 - x^(i^3)), {i, 1, m^(1/3) // Ceiling}]/ Product[1 - x^i, {i, 1, m}] + O[x]^m, x] (* _Jean-François Alcover_, Nov 14 2020 *)
%Y A264392 Cf. A264391.
%K A264392 nonn
%O A264392 0,3
%A A264392 _Emeric Deutsch_, Nov 13 2015