A036079 E.g.f.: exp((exp(p*x)-p-1)/p+exp(x)) for p=9.

1, 2, 14, 150, 1942, 29174, 505318, 9957798, 219177942, 5303780758, 139554619206, 3962202725254, 120644298135478, 3918518255860342, 135117086088186662, 4925731652244913766, 189170325211554345366, 7629758975467859662678, 322296334808561664346886

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

• 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

Mathematica

mx = 16; p = 9; 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] * 9^k * BellB[k, 1/9] * 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=9. - Vaclav Kotesovec, Jul 03 2022

a(n) ~ (9*n/LambertW(9*n))^n * exp(n/LambertW(9*n) + (9*n/LambertW(9*n))^(1/9) - n - 10/9) / sqrt(1 + LambertW(9*n)). - Vaclav Kotesovec, Jul 10 2022

Extensions

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