A261519 Expansion of Product_{k>=1} ((1+x^k)/(1-x^k))^(2^k).

1, 4, 16, 60, 208, 692, 2224, 6940, 21152, 63188, 185488, 536268, 1529648, 4310804, 12017264, 33171916, 90745472, 246201412, 662897232, 1772295020, 4707336848, 12426673188, 32617079280, 85152717404, 221183486496, 571784014244, 1471463190032, 3770577250716

Offset: 0

Author

Vaclav Kotesovec, Aug 23 2015

Keywords

• nonn

Comments

Convolution of A034899 and A102866.

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 0..1000
• Vaclav Kotesovec, Asymptotics of the Euler transform of Fibonacci numbers, arXiv:1508.01796 [math.CO]
• Vaclav Kotesovec, A method of finding the asymptotics of q-series based on the convolution of generating functions, arXiv:1509.08708 [math.CO], Sep 30 2015, p. 27.

Crossrefs

Cf. A156616, A261520, A260916, A001934, A015128.

Programs

Mathematica

nmax = 40; CoefficientList[Series[Product[((1 + x^k)/(1 - x^k))^(2^k), {k, 1, nmax}], {x, 0, nmax}], x]

Formula

a(n) ~ 2^n * exp(2*sqrt(2*n) - 1 + c) / (sqrt(Pi) * 2^(3/4) * n^(3/4)), where c = 2 * Sum_{j>=1} 1/((2*j+1)*(2^(2*j)-1)) = 0.2545212486386431009939814261118792033...