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

0,3

Comments

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

Links

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

Formula

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

Example

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.

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 *)

Crossrefs

Cf. A264391.

Sequence in context: A000070 A369579 A008609 * A100823 A102346 A333148

Adjacent sequences: A264389 A264390 A264391 * A264393 A264394 A264395

Keyword

nonn

Author

Emeric Deutsch, Nov 13 2015

Status

approved