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
KEYWORD
nonn
AUTHOR
Emeric Deutsch, Nov 13 2015
STATUS
approved