A166922 E.g.f. exp(-x)*exp(exp(2*x)/2-1/2)/2 + 1/2.

1, 0, 1, 2, 10, 48, 276, 1768, 12552, 97408, 818704, 7396384, 71380640, 732058880, 7943068992, 90833753728, 1091134058624, 13728139694080, 180436251140352, 2471790031618560

Offset: 0

Author

Karol A. Penson, Oct 23 2009

Keywords

• nonn

Comments

In general, if m >= 1, b <> 0 and e.g.f. = exp(m*exp(b*x) + r*x + s) then a(n) ~ b^n * n^(n + r/b) * exp(n/LambertW(n/m) - n + s) / (m^(r/b) * sqrt(1 + LambertW(n/m)) * LambertW(n/m)^(n + r/b)). - Vaclav Kotesovec, Jun 28 2022

Links

• G. C. Greubel, Table of n, a(n) for n = 0..250
• R. Suter, Two analogues of a classical sequence, J. Integer Sequences, Vol. 3 (2000), #P00.1.8.

Programs

Mathematica

With[{nn = 25}, CoefficientList[Series[Exp[-t]*Exp[Exp[2*t]/2 - 1/2]/2 + 1/2, {t, 0, nn}], t] Range[0, nn]!] (* G. C. Greubel, May 28 2016 *)

PARI

x='x+O('x^66); Vec(serlaplace(exp(-x)*exp(exp(2*x)/2-1/2)/2+1/2)) \\ Joerg Arndt, May 06 2013

Sage

def A166922_list(n):  # n>=1
    T = [0]*(n+1); R = [1]
    for m in (1..n-1):
        a,b,c = 1,0,0
        for k in range(m,-1,-1):
            r = a + 2*(k*(b+c)+c)
            if k < m : T[k+2] = u;
            a,b,c = T[k-1],a,b
            u = r
        T[1] = u; R.append(u/2)
    return R
A166922_list(20)
# Peter Luschny, Nov 01 2012

Formula

A004211(n) = -1 + 2*sum(k=0..n, C(n,k)*a(k)). - Peter Luschny, Nov 01 2012

G.f.: 1/2 + 1/2/Q(0), where Q(k)= 1 - 2*x*k - 2*x^2*(k+1)/Q(k+1); (continued fraction). - Sergei N. Gladkovskii, May 06 2013

a(n) ~ 2^(n - 3/2) * n^(n - 1/2) * exp(n/LambertW(2*n) - n - 1/2) / (sqrt(1 + LambertW(2*n)) * LambertW(2*n)^(n - 1/2)). - Vaclav Kotesovec, Jun 26 2022

Extensions

Definition corrected on a suggestion of M. F. Hasler, Peter Luschny, Nov 05 2012