A355233 - OEIS

%I #25 Jun 26 2022 02:58:11

%S 0,1,0,4,6,40,150,832,4494,27496,178278,1240720,9159678,71523448,

%T 588049878,5073746464,45800173038,431400176008,4230061102662,

%U 43087882883248,455079854567646,4975136823055768,56212975652894646,655496634896272960,7878552380411524302

%N E.g.f. A(x) satisfies A'(x) = 1 + 2 * (exp(x) - 1) * A(x).

%F a(0) = 0, a(1) = 1; a(n+1) = 2 * Sum_{k=1..n-1} binomial(n,k) * a(k).

%F From _Vaclav Kotesovec_, Jun 26 2022: (Start)

%F E.g.f.: 3*exp(2*exp(x) - 2*x - 2)/4 - 1/(exp(2*x)*4) - 1/(2*exp(x)).

%F a(n) = 3*A194689(n)/4 - (-1)^n * (2^(n-2) + 1/2).

%F a(n) ~ 3 * n^(n-2) * exp(n/LambertW(n/2) - n - 2) / (sqrt(1 + LambertW(n/2)) * LambertW(n/2)^(n-2)). (End)

%t nmax = 25; CoefficientList[Series[3*E^(-2 + 2*E^x - 2*x)/4 - 1/(E^(2*x)*4) - 1/(2*E^x), {x, 0, nmax}], x] * Range[0, nmax]! (* _Vaclav Kotesovec_, Jun 25 2022 *)

%o (PARI) a_vector(n) = my(v=vector(n)); v[1]=1; for(i=1, n-1, v[i+1]=2*sum(j=1, i-1, binomial(i, j)*v[j])); concat(0, v);

%Y Cf. A004123, A087650, A194689, A355206, A355232.

%K nonn

%O 0,4

%A _Seiichi Manyama_, Jun 25 2022

%E Prepended a(0)=0 from _Vaclav Kotesovec_, Jun 25 2022