A355233 E.g.f. A(x) satisfies A'(x) = 1 + 2 * (exp(x) - 1) * A(x).

0, 1, 0, 4, 6, 40, 150, 832, 4494, 27496, 178278, 1240720, 9159678, 71523448, 588049878, 5073746464, 45800173038, 431400176008, 4230061102662, 43087882883248, 455079854567646, 4975136823055768, 56212975652894646, 655496634896272960, 7878552380411524302

Offset

0,4

Links

Table of n, a(n) for n=0..24.

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

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

Sequence in context: A281223 A023644 A319672 * A145387 A034923 A013022

Adjacent sequences: A355230 A355231 A355232 * A355234 A355235 A355236

Keyword

nonn

Author

Seiichi Manyama, Jun 25 2022

Extensions

Prepended a(0)=0 from Vaclav Kotesovec, Jun 25 2022

Status

approved