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

%I A304329 #27 Mar 08 2021 08:39:22
%S A304329 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,
%T A304329 169,197,231,268,312,362,419,485,557,642,737,846,967,1108,1262,1442,
%U A304329 1640,1865,2118,2398,2719,3074,3474,3922,4421,4980,5604,6294,7070,7929
%N A304329 Expansion of Product_{k>0} (Sum_{m>=0} x^(k*m^3)).
%C A304329 Also the number of partitions of n in which each part occurs a cube number (>=0) of times.
%H A304329 Alois P. Heinz, <a href="/A304329/b304329.txt">Table of n, a(n) for n = 0..10000</a> (first 1001 terms from Seiichi Manyama)
%e A304329 n | Partitions of n in which each part occurs a cube number (>=0) of times
%e A304329 --+-----------------------------------------------------------------------
%e A304329 1 | 1;
%e A304329 2 | 2;
%e A304329 3 | 3 = 2+1;
%e A304329 4 | 4 = 3+1;
%e A304329 5 | 5 = 4+1 = 3+2;
%e A304329 6 | 6 = 5+1 = 4+2 = 3+2+1;
%e A304329 7 | 7 = 6+1 = 5+2 = 4+3 = 4+2+1;
%e A304329 8 | 8 = 7+1 = 6+2 = 5+3 = 5+2+1 = 4+3+1 = 1+1+1+1+1+1+1+1;
%p A304329 b:= proc(n, i) option remember; local j; if n=0 then 1
%p A304329       elif i<1 then 0 else b(n, i-1); for j while
%p A304329         i*j^3<=n do %+b(n-i*j^3, i-1) od; % fi
%p A304329     end:
%p A304329 a:= n-> b(n$2):
%p A304329 seq(a(n), n=0..60);  # _Alois P. Heinz_, May 11 2018
%t A304329 terms = 100;
%t A304329 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 *)
%Y A304329 Cf. A000041, A300446.
%K A304329 nonn
%O A304329 0,4
%A A304329 _Seiichi Manyama_, May 11 2018