A316143 - OEIS

%I #20 Feb 28 2024 10:47:53

%S 1,2,12,92,912,10772,148512,2328692,40842912,791302772,16767551712,

%T 385382491892,9542377300512,253105962752372,7156766466076512,

%U 214814484529608692,6819311473596695712,228212485803422931572,8028037725386962194912,296094910181041530831092

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

%C Self-convolution of A167137.

%C 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, 5, 1, 2, 6, 0, 2, 5, 1, 2, 6, 0, 2, 5, 1, 2, 6, 0, ...], with a preperiod of length 1 and an apparent period thereafter of 6 = phi(7). - _Peter Bala_, Mar 03 2023

%H Vaclav Kotesovec, <a href="/A316143/b316143.txt">Table of n, a(n) for n = 0..400</a>

%F Sum_{k=0..n} binomial(n,k) * A167137(k) * A167137(n-k).

%F a(n) ~ n! * exp(Pi * sqrt(2*n/(3*log(2))) - Pi^2 * (1 - 1/log(2)) / 12) / (2^(7/4) * 3^(3/4) * n^(5/4) * (log(2))^(n - 1/4)).

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

%Y Cf. A000712, A167137, A316142, A316144.

%K nonn,easy

%O 0,2

%A _Vaclav Kotesovec_, Jun 25 2018