A316142 Expansion of e.g.f. Product_{k>=1} (1 + (exp(x)-1)^k)^2.

1, 2, 8, 56, 476, 4832, 58508, 815936, 12750956, 220610432, 4195325708, 86976996416, 1949966347436, 46965887762432, 1208922621624908, 33111231803362496, 961354836530983916, 29490401681798152832, 952900154176192244108, 32342850619899263226176

Offset: 0

Author

Vaclav Kotesovec, Jun 25 2018

Keywords

• nonn
• easy

Comments

Self-convolution of A305550.

Conjecture: Let k be a positive integer. The sequence obtained by reducing a(n) modulo k is eventually periodic with the period dividing phi(k) = A000010(k). For example, modulo 7 we obtain the sequence [1, 2, 1, 0, 0, 2, 2, 2, 1, 0, 0, 2, 2, 2, 1, 0, 0, 2, 2, ...], with a preperiod of length 1 and an apparent period thereafter of 6 = phi(7). - Peter Bala, Mar 03 2023

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 0..400

Crossrefs

Cf. A022567, A305550, A316143, A316144.

Programs

Mathematica

nmax = 20; CoefficientList[Series[Product[(1+(Exp[x]-1)^k)^2, {k, 1, nmax}], {x, 0, nmax}], x] * Range[0, nmax]!

Formula

Sum_{k=0..n} binomial(n,k) * A305550(k) * A305550(n-k).

a(n) ~ n! * exp(Pi * sqrt(n/(3*log(2))) - Pi^2 * (1 - 1/log(2)) / 24) / (2^(5/2) * 3^(1/4) * (log(2))^(n + 1/4) * n^(3/4)).