A036078 E.g.f.: exp((exp(p*x)-p-1)/p+exp(x)) for p=8.

1, 2, 13, 127, 1508, 20859, 332557, 6019108, 121462267, 2692076295, 64846340130, 1684713690917, 46916754353013, 1393010598959594, 43889040801834505, 1461369418905803027, 51243270154712083052

Offset

0,2

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

Table of n, a(n) for n=0..16.

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

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=8. - Vaclav Kotesovec, Jul 03 2022

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

Mathematica

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 *)
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 *)

Crossrefs

Cf. A001861, A002872-A002875, A036074, ...

Sequence in context: A074365 A071362 A108471 * A290219 A057065 A259611

Adjacent sequences: A036075 A036076 A036077 * A036079 A036080 A036081

Keyword

nonn

Author

N. J. A. Sloane.

Extensions

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

Status

approved