A280451 - OEIS

%I #17 Apr 29 2021 08:44:59

%S 1,1,1,2,3,4,5,7,9,13,16,20,27,34,42,53,67,82,102,125,153,188,227,274,

%T 332,401,478,574,686,815,969,1147,1356,1600,1884,2210,2597,3040,3547,

%U 4141,4824,5607,6508,7546,8732,10100,11656,13431,15473,17793,20429,23436

%N G.f.: Product_{k>=1, j>=1} (1+x^(j*k^2)).

%H Vaclav Kotesovec, <a href="/A280451/b280451.txt">Table of n, a(n) for n = 0..10000</a>

%F a(n) ~ exp(Pi^2*sqrt(n/2)/3 + sqrt(3) * (sqrt(2)-1) * Zeta(1/2) * Zeta(3/2) * n^(1/4) / (2^(3/4) * sqrt(Pi)) - 9*((sqrt(2)-1) * Zeta(1/2) * Zeta(3/2))^2 / (16*Pi^3)) * sqrt(Pi) / (2^(3/2) * sqrt(3) * n^(3/4)).

%t nmax = 100; CoefficientList[Series[Product[(1 + x^(j*k^2)), {k, 1, Floor[Sqrt[nmax]+1]}, {j, 1, Floor[nmax/k^2] + 1}], {x, 0, nmax}], x]

%o (PARI) my(N=66, x='x+O('x^N)); Vec(prod(k=1, sqrt(N), eta(x^(2*k^2))/eta(x^(k^2)))) \\ _Seiichi Manyama_, Apr 29 2021

%Y Cf. A004101, A006171, A107742, A280663, A280664.

%K nonn

%O 0,4

%A _Vaclav Kotesovec_, Jan 03 2017