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%I #38 Feb 13 2023 08:03:14
%S 1,2,9,55,412,3619,36333,408888,5080907,68914023,1011165446,
%T 15935379409,268125052373,4792458452162,90605469012877,
%U 1805135197261131,37775862401203916,827992670793489263
%N Expansion of e.g.f. exp((exp(p*x) - p - 1)/p + exp(x)) for p=4.
%D T. S. Motzkin, Sorting numbers for cylinders and other classification numbers, in Combinatorics, Proc. Symp. Pure Math. 19, AMS, 1971, pp. 167-176.
%D T. S. Motzkin, Sorting numbers ...: for a link to an annotated scanned version of this paper see A000262.
%H Vaclav Kotesovec, <a href="https://arxiv.org/abs/2207.10568">Asymptotics for a certain group of exponential generating functions</a>, arXiv:2207.10568 [math.CO], Jul 13 2022.
%H Vladimir Kruchinin, <a href="http://arxiv.org/abs/1009.2565">Composition of ordinary generating functions</a>, arXiv:1009.2565 [math.CO], 2010.
%H <a href="/index/So#sorting">Index entries for sequences related to sorting</a>
%F a(n) = sum(sum(binomial(m,i)*sum(binomial(i,j)*(1/4)^j*(3*j+i)^n,j,0,i)*(-5/4)^(m-i),i,0,m)/m!,m,1,n), n > 0. - _Vladimir Kruchinin_, Sep 14 2010
%F a(n) ~ exp(exp(p*r)/p + exp(r) - 1 - 1/p - n) * (n/r)^(n + 1/2) / sqrt((1 + p*r)*exp(p*r) + (1 + r)*exp(r)), where r = LambertW(p*n)/p - 1/(1 + p/LambertW(p*n) + n^(1 - 1/p) * (1 + LambertW(p*n)) * (p/LambertW(p*n))^(2 - 1/p)) for p=4. - _Vaclav Kotesovec_, Jul 03 2022
%F a(n) ~ (4*n/LambertW(4*n))^n * exp(n/LambertW(4*n) + (4*n/LambertW(4*n))^(1/4) - n - 5/4) / sqrt(1 + LambertW(4*n)). - _Vaclav Kotesovec_, Jul 10 2022
%t mx = 16; p = 4; Range[0, mx]! CoefficientList[ Series[ Exp[ (Exp[p*x] - p - 1)/p + Exp[x]], {x, 0, mx}], x] (* _Robert G. Wilson v_, Dec 12 2012 *)
%t Table[Sum[Binomial[n,k] * 4^k * BellB[k, 1/4] * BellB[n-k], {k, 0, n}], {n, 0, 20}] (* _Vaclav Kotesovec_, Jun 29 2022 *)
%o (Maxima) a(n):=sum(sum(binomial(m,i)*sum(binomial(i,j)*(1/4)^j*(3*j+i)^n,j,0,i)*(-5/4)^(m-i),i,0,m)/m!,m,1,n); /* _Vladimir Kruchinin_, Sep 14 2010 */
%Y Cf. A001861, A002872, A002873, A002874, A002875, A036076, ...
%K nonn
%O 0,2
%A _N. J. A. Sloane_
%E Edited by _N. J. A. Sloane_, Jul 11 2008 at the suggestion of Franklin T. Adams-Watters
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