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%I #22 Jul 22 2022 02:19:27
%S 1,2,13,127,1508,20859,332557,6019108,121462267,2692076295,
%T 64846340130,1684713690917,46916754353013,1393010598959594,
%U 43889040801834505,1461369418905803027,51243270154712083052
%N E.g.f.: exp((exp(p*x)-p-1)/p+exp(x)) for p=8.
%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=8. - _Vaclav Kotesovec_, Jul 03 2022
%F a(n) ~ (8*n/LambertW(8*n))^n * exp(n/LambertW(8*n) + (8*n/LambertW(8*n))^(1/8) - n - 9/8) / sqrt(1 + LambertW(8*n)). - _Vaclav Kotesovec_, Jul 10 2022
%t mx = 16; p = 8; 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] * 8^k * BellB[k, 1/8] * BellB[n-k], {k, 0, n}], {n, 0, 20}] (* _Vaclav Kotesovec_, Jun 29 2022 *)
%Y Cf. A001861, A002872-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.
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