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%I #28 Nov 28 2022 02:07:26
%S 1,2,11,87,844,9599,125545,1854234,30407763,546409567,10654642428,
%T 223763443039,5030118977041,120393730088818,3054106291046267,
%U 81792080931311015,2304639285452820684,68117438479292896255
%N Expansion of e.g.f. exp((exp(p*x)-p-1)/p+exp(x)) for p=6.
%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 Robert Israel, <a href="/A036076/b036076.txt">Table of n, a(n) for n = 0..437</a>
%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=6. - _Vaclav Kotesovec_, Jul 03 2022
%F a(n) ~ (6*n/LambertW(6*n))^n * exp(n/LambertW(6*n) + (6*n/LambertW(6*n))^(1/6) - n - 7/6) / sqrt(1 + LambertW(6*n)). - _Vaclav Kotesovec_, Jul 10 2022
%p egf:= exp((exp(6*x)-6-1)/6+exp(x)):
%p S:= series(egf,x,501):
%p seq(coeff(S,x,i)*i!, i=0..20); # _Robert Israel_, Nov 27 2022
%t mx = 16; p = 6; 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] * 6^k * BellB[k, 1/6] * 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.
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