|
%I #8 Jan 07 2017 10:57:10
%S 1,1,1,2,2,3,4,5,6,8,10,12,15,18,22,27,33,39,47,56,66,79,93,109,128,
%T 150,175,204,237,274,318,367,423,487,559,641,734,839,957,1091,1241,
%U 1410,1601,1814,2053,2322,2622,2957,3334,3752,4218,4740,5318,5962,6679
%N G.f.: Product_{k>=1, j>=1} (1 + x^(j*k^4)).
%C In general, if m>=3 and g.f. = Product_{k>=1, j>=1} (1+x^(j*k^m)), then a(n, m) ~ exp(Pi*sqrt(Zeta(m)*n/3) + (2^(1/m)-1) * Pi^(-1/m) * Gamma(1+1/m) * Zeta(1+1/m) * Zeta(1/m) * (3*n/Zeta(m))^(1/(2*m))) * Zeta(m)^(1/4) / (2^(5/4) * 3^(1/4) * n^(3/4)).
%H Vaclav Kotesovec, <a href="/A280664/b280664.txt">Table of n, a(n) for n = 0..10000</a>
%F a(n) ~ exp(Pi^3 * sqrt(n/30)/3 + 2^(-15/8) * 3^(3/8) * 5^(1/8) * (2^(1/4)-1) * Pi^(-3/4) * Gamma(1/4) * Zeta(5/4) * Zeta(1/4) * n^(1/8)) * Pi / (2^(3/2) * 3^(3/4) * 5^(1/4) * n^(3/4)).
%t nmax = 100; CoefficientList[Series[Product[1+x^(j*k^4), {k, 1, Floor[nmax^(1/4)]+1}, {j, 1, Floor[nmax/k^4]+1}], {x, 0, nmax}], x]
%Y Cf. A107742 (m=1), A280451 (m=2), A280663 (m=3).
%Y Cf. A280662.
%K nonn
%O 0,4
%A _Vaclav Kotesovec_, Jan 07 2017
|
