%I #24 Jul 22 2022 02:18:30
%S 1,2,14,150,1942,29174,505318,9957798,219177942,5303780758,
%T 139554619206,3962202725254,120644298135478,3918518255860342,
%U 135117086088186662,4925731652244913766,189170325211554345366,7629758975467859662678,322296334808561664346886
%N E.g.f.: exp((exp(p*x)-p-1)/p+exp(x)) for p=9.
%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 <a href="/index/So#sorting">Index entries for sequences related to sorting</a>
%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=9. - _Vaclav Kotesovec_, Jul 03 2022
%F a(n) ~ (9*n/LambertW(9*n))^n * exp(n/LambertW(9*n) + (9*n/LambertW(9*n))^(1/9) - n - 10/9) / sqrt(1 + LambertW(9*n)). - _Vaclav Kotesovec_, Jul 10 2022
%t mx = 16; p = 9; 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] * 9^k * BellB[k, 1/9] * BellB[n-k], {k, 0, n}], {n, 0, 20}] (* _Vaclav Kotesovec_, Jun 29 2022 *)
%Y Cf. A001861, A002872, A002873, A002874, A002875, A036074...
%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_.