A377080 - OEIS

%I #10 Oct 15 2024 11:52:09

%S 0,0,1,1,0,0,0,0,1,1,1,1,0,0,0,0,0,0,1,1,1,2,1,1,1,0,0,0,0,0,0,0,1,1,

%T 1,2,2,2,2,2,1,1,1,0,0,0,0,0,0,0,1,1,1,2,2,3,3,3,3,3,3,2,2,1,1,1,0,0,

%U 0,0,0,0,1,1,1,2,2,3,4,4,4,5,5,5,5,4,4

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

%H Vaclav Kotesovec, <a href="/A377080/b377080.txt">Table of n, a(n) for n = 0..10000</a>

%F a(n) ~ (1+r) * exp(sqrt((40*log(r)^2 + 4*polylog(2, 1/(1+r)) - Pi^2/3)*n)) / (2*sqrt((4 + 5*r)*n)), where r = A230152 = 0.856674883854502874852324... is the real root of the equation r^4*(1+r) = 1.

%t nmax = 150; CoefficientList[Series[Sum[x^(2*k^2)*Product[1+x^j, {j, 1, k}], {k, 1, Sqrt[nmax/2]}], {x, 0, nmax}], x]

%Y Cf. A306734, A350890, A350893, A377081.

%Y Cf. A230152.

%K nonn

%O 0,22

%A _Vaclav Kotesovec_, Oct 15 2024