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%I #15 Oct 13 2020 23:51:12
%S 1,1,7,57,607,8121,130527,2447257,52435327,1263925881,33851510047,
%T 997303255257,32052722909247,1116001351341241,41845542004844767,
%U 1681112968022124057,72039936723424794367,3280036569708658302201,158127582939120607830687,8046697501049910668173657
%N a(0) = 1; a(n) = (1/3) * Sum_{k=1..n} binomial(n,k) * (4^k-1) * a(n-k).
%H Robert Israel, <a href="/A337556/b337556.txt">Table of n, a(n) for n = 0..387</a>
%F E.g.f.: 3 / (3 + exp(x) - exp(4*x)).
%F a(n) ~ n! * (r^3 - 1) * (4*r^3 - 16*r^2 + 64*r - 3) / (771 * log(r)^(n+1)), where r = 1.452626878833844... is the positive real root of the equation r*(r^3 - 1) = 3. - _Vaclav Kotesovec_, Aug 31 2020
%p E:= 3 / (3 + exp(x) - exp(4*x)):
%p S:= series(E,x,41):
%p seq(n!*coeff(S,x,n),n=0..40); # _Robert Israel_, Oct 13 2020
%t a[0] = 1; a[n_] := a[n] = (1/3) Sum[Binomial[n, k] (4^k - 1) a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 19}]
%t nmax = 19; CoefficientList[Series[3/(3 + Exp[x] - Exp[4 x]), {x, 0, nmax}], x] Range[0, nmax]!
%o (PARI) seq(n)={Vec(serlaplace(3 / (3 + exp(x + O(x*x^n)) - exp(4*x + O(x*x^n)))))} \\ _Andrew Howroyd_, Aug 31 2020
%Y Cf. A000556, A002450, A337555.
%K nonn
%O 0,3
%A _Ilya Gutkovskiy_, Aug 31 2020