%I #25 Jul 22 2022 02:14:52

%S 1,2,17,231,3724,68819,1464781,35645040,973624491,29313919207,

%T 960689482494,33997330377817,1291521482389621,52395164853506674,

%U 2259005857941805253,103064324686839195035,4957382457319437575820,250592665906288206715951,13275467282249493427541201

%N E.g.f.: exp((exp(p*x)-p-1)/p+exp(x)) for p=12.

%C In general, for p>=2, a(n) ~ c * (p*n/LambertW(p*n))^n * exp(n/LambertW(p*n) + (p*n/LambertW(p*n))^(1/p) - n - 1 - 1/p) / sqrt(1 + LambertW(p*n)), where c = 1 for p>=3 and c = exp(-1/4) for p=2. - _Vaclav Kotesovec_, Jul 10 2022

%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=12. - _Vaclav Kotesovec_, Jul 03 2022

%F a(n) ~ (12*n/LambertW(12*n))^n * exp(n/LambertW(12*n) + (12*n/LambertW(12*n))^(1/12) - n - 13/12) / sqrt(1 + LambertW(12*n)). - _Vaclav Kotesovec_, Jul 10 2022

%t mx = 16; p = 12; 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] * 12^k * BellB[k, 1/12] * 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_.