Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).
%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