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Expansion of Product_{k>=1} 1/(1 - x^((2*k-1)^3)).
4

%I #12 Sep 19 2017 05:43:24

%S 1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,

%T 2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,

%U 3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5

%N Expansion of Product_{k>=1} 1/(1 - x^((2*k-1)^3)).

%C Number of partitions of n into odd cubes.

%C In general, if m > 0 and g.f. = Product_{k>=1} 1/(1 - x^((2*k-1)^m)), then a(n) ~ exp((m+1) * (Gamma(1/m) * Zeta(1+1/m) / (2*m^2))^(m/(m+1)) * n^(1/(m+1))) * (Gamma(1/m) * Zeta(1+1/m))^(m/(2*(m+1))) / (sqrt(Pi*(m+1)) * 2^((1+m*(m+3))/(2*(m+1))) * m^((m-1)/(2*(m+1))) * n^((2*m+1)/(2*(m+1)))). - _Vaclav Kotesovec_, Sep 19 2017

%H Vaclav Kotesovec, <a href="/A287091/b287091.txt">Table of n, a(n) for n = 0..10000</a>

%H <a href="/index/Su#ssq">Index entries for sequences related to sums of cubes</a>

%H <a href="/index/Par#part">Index entries for sequences related to partitions</a>

%F G.f.: Product_{k>=1} 1/(1 - x^((2*k-1)^3)).

%F a(n) ~ exp(2^(5/4) * (Gamma(1/3) * Zeta(4/3))^(3/4) * n^(1/4) / 3^(3/2)) * (Gamma(1/3) * Zeta(4/3)/2)^(3/8) / (8 * 3^(1/4) * sqrt(Pi) * n^(7/8)). - _Vaclav Kotesovec_, Sep 18 2017

%e a(27) = 2 because we have [27] and [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1].

%t nmax = 110; CoefficientList[Series[Product[1/(1 - x^((2*k-1)^3)), {k, 1, Floor[nmax^(1/3)/2] + 1}], {x, 0, nmax}], x] (* _Vaclav Kotesovec_, Sep 18 2017 *)

%Y Cf. A000009 (m=1), A167661 (m=2).

%Y Cf. A003108, A016755, A259792, A279329, A280865.

%Y Cf. A001156, A046042, A292547.

%K nonn

%O 0,28

%A _Ilya Gutkovskiy_, May 19 2017