A349560 E.g.f. satisfies log(A(x)) = (exp(x*A(x)) - 1) * x.

1, 0, 2, 3, 40, 245, 2976, 35287, 524560, 8790777, 165530800, 3493679651, 80812685064, 2049413147509, 56294089065592, 1668771901644135, 53057068616526496, 1801519375618579313, 65063987978980974048, 2490449984485154892235, 100716775979173952155480

Offset

0,3

Links

Table of n, a(n) for n=0..20.

Formula

a(n) ~ sqrt(s*(1 - r^2*s/(1 + r*s))) * n^(n-1) / (exp(n) * r^(n + 1/2)), where r = 0.4599551063707173872728335298048828687860291021728... is the root of the equation r - LambertW(1/r) - 2*log(r) = 1/LambertW(1/r) and s = LambertW(1/r)/r = 1.938208283387405345404104769972407921289092368509... - Vaclav Kotesovec, Nov 22 2021

a(n) = n! * Sum_{k=0..floor(n/2)} (n-k+1)^(k-1) * Stirling2(n-k,k)/(n-k)!. - Seiichi Manyama, Aug 27 2022

Maple

a:= n-> n!*coeff(series(RootOf(A=exp(x*exp(x*A)-x), A), x, n+1), x, n):
seq(a(n), n=0..20);  # Alois P. Heinz, Nov 22 2021

Mathematica

nmax = 20; A[_] = 0; Do[A[x_] = Exp[(E^(x*A[x]) - 1)*x] + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x] * Range[0, nmax]! (* Vaclav Kotesovec, Nov 22 2021 *)

Prog

(PARI) my(A=1, n=22); for(i=1, n, A=exp((exp(x*A)-1)*(x+x*O(x^n)))); Vec(serlaplace(A))
(PARI) a(n) = n!*sum(k=0, n\2, (n-k+1)^(k-1)*stirling(n-k, k, 2)/(n-k)!); \\ Seiichi Manyama, Aug 27 2022

Crossrefs

Cf. A052506, A349557.

Cf. A356785, A356788, A356789.

Sequence in context: A097170 A231797 A355235 * A176366 A157132 A262189

Adjacent sequences: A349557 A349558 A349559 * A349561 A349562 A349563

Keyword

nonn

Author

Seiichi Manyama, Nov 22 2021

Status

approved