A394205 - OEIS

%I #8 Mar 13 2026 10:26:30

%S 1,0,2,0,2,0,2,2,2,4,2,4,2,4,4,6,6,8,6,8,6,10,10,12,14,12,16,12,20,16,

%T 24,20,26,24,26,30,30,36,34,40,40,44,48,52,54,58,58,68,66,78,78,84,88,

%U 88,102,100,116,118,126,132,132,148,150,166,174,180,194,192

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

%C Convolution of A296238 and A296237.

%H Vaclav Kotesovec, <a href="/A394205/b394205.txt">Table of n, a(n) for n = 0..10000</a>

%F a(n) ~ Gamma(1 + b/d) * ((4-sqrt(2))*zeta(3/2))^(2/3 + b/(3*d)) * d^(1/6 + b/(3*d)) * exp(3*Pi^(1/3) * ((4-sqrt(2))*zeta(3/2))^(2/3) * (n/d)^(1/3) / 4) / (2^(7/2 + 3*b/(2*d)) * sqrt(3) * Pi^(7/6 - b/(6*d)) * n^(7/6 + b/(3*d))), where d = 3/2, b = 1/2.

%t nmax = 120; CoefficientList[Series[Product[(1 + x^(k*(3*k+1)/2)) / (1 - x^(k*(3*k+1)/2)), {k, 1, Floor[Sqrt[1 + 24*nmax]/6 + 1]}], {x, 0, nmax}], x]

%Y Cf. A103265, A296238, A296237.

%K nonn

%O 0,3

%A _Vaclav Kotesovec_, Mar 12 2026