A036076 Expansion of e.g.f. exp((exp(p*x)-p-1)/p+exp(x)) for p=6.

1, 2, 11, 87, 844, 9599, 125545, 1854234, 30407763, 546409567, 10654642428, 223763443039, 5030118977041, 120393730088818, 3054106291046267, 81792080931311015, 2304639285452820684, 68117438479292896255

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn

References

• T. S. Motzkin, Sorting numbers for cylinders and other classification numbers, in Combinatorics, Proc. Symp. Pure Math. 19, AMS, 1971, pp. 167-176.
• T. S. Motzkin, Sorting numbers ...: for a link to an annotated scanned version of this paper see A000262.

Links

• Robert Israel, Table of n, a(n) for n = 0..437
• Vaclav Kotesovec, Asymptotics for a certain group of exponential generating functions, arXiv:2207.10568 [math.CO], Jul 13 2022.
• Index entries for sequences related to sorting

Crossrefs

Cf. A001861, A002872, A002873, A002874, A002875, A036074, ...

Programs

Maple

egf:=  exp((exp(6*x)-6-1)/6+exp(x)):
S:= series(egf,x,501):
seq(coeff(S,x,i)*i!, i=0..20); # Robert Israel, Nov 27 2022

Mathematica

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 *)
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 *)

Formula

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

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

Extensions

Edited by N. J. A. Sloane, Jul 11 2008 at the suggestion of Franklin T. Adams-Watters.