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A036080 E.g.f.: exp((exp(p*x)-p-1)/p+exp(x)) for p=10.

%I A036080 #26 Jul 08 2025 21:34:41
%S A036080 1,2,15,175,2452,39703,741177,15771270,375485507,9837064575,
%T A036080 280338965720,8623355105347,284589703065137,10022926411599482,
%U A036080 374900187362983015,14830483377507515247,618219446355189917804,27071966121397255354079,1241912851303663452150377
%N A036080 E.g.f.: exp((exp(p*x)-p-1)/p+exp(x)) for p=10.
%D A036080 T. S. Motzkin, Sorting numbers for cylinders and other classification numbers, in Combinatorics, Proc. Symp. Pure Math. 19, AMS, 1971, pp. 167-176.
%D A036080 T. S. Motzkin, Sorting numbers ...: for a link to an annotated scanned version of this paper see A000262.
%H A036080 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 A036080 <a href="/index/So#sorting">Index entries for sequences related to sorting</a>
%F A036080 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=10. - _Vaclav Kotesovec_, Jul 03 2022
%F A036080 a(n) ~ (10*n/LambertW(10*n))^n * exp(n/LambertW(10*n) + (10*n/LambertW(10*n))^(1/10) - n - 11/10) / sqrt(1 + LambertW(10*n)). - _Vaclav Kotesovec_, Jul 10 2022
%t A036080 mx = 16; p = 10; 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 A036080 Table[Sum[Binomial[n,k] * 10^k * BellB[k, 1/10] * BellB[n-k], {k, 0, n}], {n, 0, 20}] (* _Vaclav Kotesovec_, Jun 29 2022 *)
%Y A036080 Cf. A001861, A002872, A002873, A002874, A002875, A036074, ...
%K A036080 nonn
%O A036080 0,2
%A A036080 _N. J. A. Sloane_
%E A036080 Edited by _N. J. A. Sloane_, Jul 11 2008 at the suggestion of _Franklin T. Adams-Watters_.