A376947 G.f.: Sum_{k>=0} 2^k * x^(k*(k+1)) / Product_{j=1..k} (1 - x^j).

1, 0, 2, 2, 2, 2, 6, 6, 10, 10, 14, 14, 26, 26, 38, 46, 58, 66, 86, 94, 130, 146, 182, 214, 274, 306, 382, 438, 530, 602, 750, 838, 1018, 1162, 1390, 1598, 1898, 2154, 2550, 2910, 3402, 3858, 4550, 5134, 5970, 6786, 7846, 8902, 10306, 11618, 13390, 15142, 17346, 19562, 22398

Offset: 0

Author

Vaclav Kotesovec, Oct 10 2024

Keywords

• nonn

Comments

In general, if d >= 1, b > 0 and g.f. = Sum_{k>=0} d^k * x^(b*k^2 + c*k) / Product_{j=1..k} (1 - x^j), then a(n) ~ r^c * (b*log(r)^2 + polylog(2, 1-r))^(1/4) * exp(2*sqrt((b*log(r)^2 + polylog(2, 1-r))*n)) / (2*sqrt((2*b*(1-r) + r)*Pi) * n^(3/4)), where r is the smallest positive real root of the equation d*r^(2*b) + r = 1.

Links

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

Crossrefs

Cf. A003106, A032302, A376943, A376948.

Programs

Mathematica

nmax = 80; CoefficientList[Series[Sum[2^k*x^(k*(k+1)) / Product[1-x^j, {j, 1, k}], {k, 0, Sqrt[nmax]}], {x, 0, nmax}], x]

Formula

a(n) ~ (Pi^2/6 + log(2)^2)^(1/4) * exp(sqrt((Pi^2/3 + 2*log(2)^2)*n)) / (2^(7/4) * sqrt(3*Pi) * n^(3/4)).