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%I A338040 #15 Jun 02 2022 04:34:40
%S A338040 1,4,132,11140,1763076,449262724,168055179012,86720706877060,
%T A338040 59029852191779076,51241585497612147844,55245853646893977682692,
%U A338040 72423868722672448652558980,113447698393867318106045295876,209271794145089904620369489016964
%N A338040 E.g.f.: Sum_{j>=0} 4^j * (exp(j*x) - 1)^j.
%C A338040 In general, if k > 0 and e.g.f.: Sum_{j>=0} k^j * (exp(j*x) - 1)^j, then a(n) ~ c * (1 + k*exp(1/r))^n * r^(2*n) * n!^2 / sqrt(n), where r is the root of the equation exp(1/r) * (1 + 1/(r*LambertW(-exp(-1/r)/r))) = -1/k and c is a constant (dependent only on k).
%C A338040 Conjecture: Let p be prime. The sequence obtained by reducing a(n) modulo p for n >= 1 is purely periodic with period p - 1. For example, modulo 7 the sequence becomes [4, 6, 3, 0, 1, 0, 4, 6, 3, 0, 1, 0, 4, 6, 3, 0, 1, 0, ...], with an apparent period of 6. - _Peter Bala_, May 31 2022
%H A338040 Seiichi Manyama, <a href="/A338040/b338040.txt">Table of n, a(n) for n = 0..207</a>
%F A338040 a(n) = Sum_{j=0..n} 4^j * j^n * j! * Stirling2(n,j).
%F A338040 a(n) ~ c * (1 + 4*exp(1/r))^n * r^(2*n) * n!^2 / sqrt(n), where r = 0.95894043087329419322124137165060249611787608513866855417024... is the root of the equation exp(1/r) * (1 + 1/(r*LambertW(-exp(-1/r)/r))) = -1/4 and c = 0.37483929689722634406486945426531890297038414869116425498643733178324...
%t A338040 Flatten[{1, Table[Sum[4^j * j^n * j! * StirlingS2[n, j], {j, 0, n}], {n, 1, 20}]}]
%t A338040 nmax = 20; CoefficientList[Series[1 + Sum[4^j*(Exp[j*x] - 1)^j, {j, 1, nmax}], {x, 0, nmax}], x] * Range[0, nmax]!
%o A338040 (PARI) {a(n)=local(X=x+x*O(x^n)); n!*polcoeff(sum(m=0, n, 4^m*(exp(m*X)-1)^m), n)}
%Y A338040 Cf. A122399, A195005, A195263, A195415, A220181, A221077, A221078, A224899, A245322.
%Y A338040 Cf. A122400, A301581, A301582, A301583.
%K A338040 nonn
%O A338040 0,2
%A A338040 _Vaclav Kotesovec_, Oct 08 2020

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