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%I #22 Feb 28 2024 10:47:48
%S 1,2,8,56,476,4832,58508,815936,12750956,220610432,4195325708,
%T 86976996416,1949966347436,46965887762432,1208922621624908,
%U 33111231803362496,961354836530983916,29490401681798152832,952900154176192244108,32342850619899263226176
%N Expansion of e.g.f. Product_{k>=1} (1 + (exp(x)-1)^k)^2.
%C Self-convolution of A305550.
%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, 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
%H Vaclav Kotesovec, <a href="/A316142/b316142.txt">Table of n, a(n) for n = 0..400</a>
%F Sum_{k=0..n} binomial(n,k) * A305550(k) * A305550(n-k).
%F 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)).
%t nmax = 20; CoefficientList[Series[Product[(1+(Exp[x]-1)^k)^2, {k, 1, nmax}], {x, 0, nmax}], x] * Range[0, nmax]!
%Y Cf. A022567, A305550, A316143, A316144.
%K nonn,easy
%O 0,2
%A _Vaclav Kotesovec_, Jun 25 2018
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