A367891 - OEIS

%I #8 Dec 05 2023 04:26:18

%S 1,0,4,4,52,164,1364,7620,60148,449252,3831700,33811716,320082228,

%T 3178774564,33234163668,363535920196,4153091085172,49406896240996,

%U 610777358429204,7830140410294148,103914148870277556,1425254885630973604,20173671034640405588

%N Expansion of e.g.f. exp(4*(exp(x) - 1 - x)).

%F G.f. A(x) satisfies: A(x) = 1 - 4 * x * ( A(x) - A(x/(1 - x)) / (1 - x) ).

%F a(n) = exp(-4) * Sum_{k>=0} 4^k * (k-4)^n / k!.

%F a(0) = 1; a(n) = 4 * Sum_{k=1..n-1} binomial(n-1,k) * a(n-k-1).

%F a(n) = Sum_{k=0..n} binomial(n,k) * (-4)^(n-k) * A078944(k).

%t nmax = 22; CoefficientList[Series[Exp[4 (Exp[x] - 1 - x)], {x, 0, nmax}], x] Range[0, nmax]!

%t a[0] = 1; a[n_] := a[n] = 4 Sum[Binomial[n - 1, k] a[n - k - 1], {k, 1, n - 1}]; Table[a[n], {n, 0, 22}]

%t Table[Sum[Binomial[n, k] (-4)^(n - k) BellB[k, 4], {k, 0, n}], {n, 0, 22}]

%o (PARI) my(x='x+O('x^30)); Vec(serlaplace(exp(4*(exp(x) - 1 - x)))) \\ _Michel Marcus_, Dec 04 2023

%Y Cf. A000296, A078944, A194689, A346739, A367890.

%K nonn

%O 0,3

%A _Ilya Gutkovskiy_, Dec 04 2023