A376944 G.f.: Sum_{k>=0} 2^k * x^(k*(k+1)/2) * Product_{j=1..k} (1 + x^j).

1, 2, 2, 4, 4, 4, 12, 8, 8, 16, 24, 24, 24, 32, 32, 64, 64, 64, 80, 80, 112, 160, 160, 160, 224, 224, 256, 320, 416, 416, 480, 576, 576, 704, 768, 896, 1152, 1216, 1280, 1536, 1600, 1856, 2112, 2304, 2560, 3200, 3456, 3584, 4224, 4480, 5120, 5760, 6144, 6656, 7808, 9088

Offset: 0

Author

Vaclav Kotesovec, Oct 10 2024

Keywords

• nonn

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 0..10000

Crossrefs

Cf. A032302, A053261, A376943, A376945.

Programs

Mathematica

nmax = 60; CoefficientList[Series[Sum[2^k * x^(k*(k+1)/2) * Product[1+x^j, {j, 1, k}], {k, 0, Sqrt[2*nmax]}], {x, 0, nmax}], x]
nmax = 60; p = 1; s = 1; Do[p = Normal[Series[2*p*(1 + x^k) * x^k, {x, 0, nmax}]]; s += p; , {k, 1, Sqrt[2*nmax]}]; Take[CoefficientList[s, x], nmax + 1]

Formula

a(n) ~ sqrt(1 + sqrt(3)) * exp(sqrt((2*log(2)^2 + 2*log(1 - sqrt(3)/2) * log(sqrt(3) - 1) + 4*polylog(2, sqrt(3) - 1) - Pi^2/3)*n)) / (4*3^(1/4)*sqrt(n)).