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A355410 Expansion of e.g.f. 1/(3 - exp(x) - exp(3*x)).

%I A355410 #12 Dec 04 2023 06:36:06
%S A355410 1,4,42,652,13482,348484,10809282,391162972,16177467642,752689508404,
%T A355410 38911563009522,2212759299753292,137270821971529002,
%U A355410 9225382887659221924,667690580181890112162,51776098497454677943612,4282645413209764715753562
%N A355410 Expansion of e.g.f. 1/(3 - exp(x) - exp(3*x)).
%F A355410 a(0) = 1; a(n) = Sum_{k=1..n} (3^k + 1) * binomial(n,k) * a(n-k).
%F A355410 a(n) ~ n! / ((9 - 2*r) * log(r)^(n+1)), where r = -2*sinh(log((-9*sqrt(3) + sqrt(247))/2)/3)/sqrt(3). - _Vaclav Kotesovec_, Jul 01 2022
%p A355410 A355410 := proc(n)
%p A355410     option remember ;
%p A355410     if n = 0 then
%p A355410         1;
%p A355410     else
%p A355410         add((3^k + 1) * binomial(n,k) * procname(n-k),k=1..n) ;
%p A355410     end if;
%p A355410 end proc:
%p A355410 seq(A355410(n),n=0..70) ; # _R. J. Mathar_, Dec 04 2023
%o A355410 (PARI) my(N=20, x='x+O('x^N)); Vec(serlaplace(1/(3-exp(x)-exp(3*x))))
%o A355410 (PARI) a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=sum(j=1, i, (3^j+1)*binomial(i, j)*v[i-j+1])); v;
%Y A355410 Cf. A004700, A355408.
%K A355410 nonn
%O A355410 0,2
%A A355410 _Seiichi Manyama_, Jul 01 2022