A304329 Expansion of Product_{k>0} (Sum_{m>=0} x^(k*m^3)).

1, 1, 1, 2, 2, 3, 4, 5, 7, 8, 11, 13, 16, 20, 24, 30, 36, 44, 51, 62, 74, 88, 103, 122, 145, 169, 197, 231, 268, 312, 362, 419, 485, 557, 642, 737, 846, 967, 1108, 1262, 1442, 1640, 1865, 2118, 2398, 2719, 3074, 3474, 3922, 4421, 4980, 5604, 6294, 7070, 7929

Offset: 0

Author

Seiichi Manyama, May 11 2018

Keywords

• nonn

Comments

Also the number of partitions of n in which each part occurs a cube number (>=0) of times.

Examples

			n | Partitions of n in which each part occurs a cube number (>=0) of times
--+-----------------------------------------------------------------------
1 | 1;
2 | 2;
3 | 3 = 2+1;
4 | 4 = 3+1;
5 | 5 = 4+1 = 3+2;
6 | 6 = 5+1 = 4+2 = 3+2+1;
7 | 7 = 6+1 = 5+2 = 4+3 = 4+2+1;
8 | 8 = 7+1 = 6+2 = 5+3 = 5+2+1 = 4+3+1 = 1+1+1+1+1+1+1+1;
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..10000 (first 1001 terms from Seiichi Manyama)

Crossrefs

Cf. A000041, A300446.

Programs

Maple

b:= proc(n, i) option remember; local j; if n=0 then 1
      elif i<1 then 0 else b(n, i-1); for j while
        i*j^3<=n do %+b(n-i*j^3, i-1) od; % fi
    end:
a:= n-> b(n$2):
seq(a(n), n=0..60);  # Alois P. Heinz, May 11 2018

Mathematica

terms = 100;
Product[Sum[x^(k*m^3), {m, 0, Ceiling[terms^(1/3)]}], {k, 1, terms}] + O[x]^terms // CoefficientList[#, x]& (* Jean-François Alcover, Mar 08 2021 *)