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

0, 1, 2, 4, 7, 12, 19, 30, 46, 68, 99, 142, 200, 279, 384, 523, 707, 946, 1256, 1656, 2169, 2822, 3652, 4699, 6017, 7666, 9725, 12282, 15452, 19362, 24176, 30080, 37307, 46117, 56843, 69854, 85613, 104640, 127578, 155150, 188249, 227872, 275242, 331738, 399027, 478988

Offset: 0

Author

Emeric Deutsch, Nov 13 2015

Keywords

• nonn

Comments

a(n) = Sum_{k=0..n} k*A264391(n,k).

Examples

			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.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..1000

Crossrefs

Cf. A264391.

Programs

Maple

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);

Mathematica

cnt[P_List] := Count[P, p_ /; IntegerQ[p^(1/3)]];
a[n_] := a[n] = cnt /@ IntegerPartitions[n] // Total;
Table[Print[n, " ", a[n]]; a[n], {n, 0, 50}];
(* or: *)
m = 50;
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 *)

Formula

G.f.: ( Sum_{i>0} x^(h(i))/(1-x^(h(i))) ) / ( Product_{i>0} 1-x^i ), where h(i) = i^3.