A280263 - OEIS

%I #7 Dec 30 2016 09:47:27

%S 1,2,2,2,2,2,2,2,4,6,6,6,6,6,6,6,8,10,10,10,10,10,10,10,12,14,14,16,

%T 18,18,18,18,20,22,22,26,30,30,30,30,32,34,34,38,42,42,42,42,44,46,46,

%U 50,54,54,56,58,60,62,62,66,70,70,74,78,82,86,86,90,94

%N G.f.: Product_{k>=1} (1+x^(k^3)) / (1-x^(k^3)).

%C Convolution of A003108 and A279329.

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

%H Vaclav Kotesovec, <a href="/A280263/b280263.txt">Table of n, a(n) for n = 0..10000</a>

%F a(n) ~ exp(2^(7/4) * ((2^(4/3)-1) * Gamma(1/3) * Zeta(4/3))^(3/4) * n^(1/4) / 3^(3/2)) * ((2^(4/3)-1) * Gamma(1/3) * Zeta(4/3))^(3/4) / (3 * 2^(15/4) * Pi^2 * n^(5/4)).

%t nmax=150; CoefficientList[Series[Product[(1+x^(k^3))/(1-x^(k^3)), {k, 1, nmax}], {x, 0, nmax}], x]

%Y Cf. A003108, A015128, A103265, A279329.

%K nonn

%O 0,2

%A _Vaclav Kotesovec_, Dec 30 2016