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E.g.f. exp(-x)*exp(exp(2*x)/2-1/2)/2 + 1/2.
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%I #23 Jun 28 2022 06:46:29

%S 1,0,1,2,10,48,276,1768,12552,97408,818704,7396384,71380640,732058880,

%T 7943068992,90833753728,1091134058624,13728139694080,180436251140352,

%U 2471790031618560

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

%C 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

%H G. C. Greubel, <a href="/A166922/b166922.txt">Table of n, a(n) for n = 0..250</a>

%H R. Suter, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL3/SUTER/sut1.html">Two analogues of a classical sequence</a>, J. Integer Sequences, Vol. 3 (2000), #P00.1.8.

%F A004211(n) = -1 + 2*sum(k=0..n, C(n,k)*a(k)). - _Peter Luschny_, Nov 01 2012

%F 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

%F 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

%t 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 *)

%o (Sage)

%o def A166922_list(n): # n>=1

%o T = [0]*(n+1); R = [1]

%o for m in (1..n-1):

%o a,b,c = 1,0,0

%o for k in range(m,-1,-1):

%o r = a + 2*(k*(b+c)+c)

%o if k < m : T[k+2] = u;

%o a,b,c = T[k-1],a,b

%o u = r

%o T[1] = u; R.append(u/2)

%o return R

%o A166922_list(20)

%o # _Peter Luschny_, Nov 01 2012

%o (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

%K nonn

%O 0,4

%A _Karol A. Penson_, Oct 23 2009

%E Definition corrected on a suggestion of _M. F. Hasler_, _Peter Luschny_, Nov 05 2012