OFFSET
0,4
FORMULA
a(0) = 0, a(1) = 1; a(n+1) = 2 * Sum_{k=1..n-1} binomial(n,k) * a(k).
From Vaclav Kotesovec, Jun 26 2022: (Start)
E.g.f.: 3*exp(2*exp(x) - 2*x - 2)/4 - 1/(exp(2*x)*4) - 1/(2*exp(x)).
a(n) = 3*A194689(n)/4 - (-1)^n * (2^(n-2) + 1/2).
a(n) ~ 3 * n^(n-2) * exp(n/LambertW(n/2) - n - 2) / (sqrt(1 + LambertW(n/2)) * LambertW(n/2)^(n-2)). (End)
MATHEMATICA
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 *)
PROG
(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);
CROSSREFS
KEYWORD
nonn
AUTHOR
Seiichi Manyama, Jun 25 2022
EXTENSIONS
Prepended a(0)=0 from Vaclav Kotesovec, Jun 25 2022
STATUS
approved