OFFSET
0,4
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.
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
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 *)
PROG
(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
(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
CROSSREFS
KEYWORD
nonn
AUTHOR
Karol A. Penson, Oct 23 2009
EXTENSIONS
Definition corrected on a suggestion of M. F. Hasler, Peter Luschny, Nov 05 2012
STATUS
approved