A007832 Number of point labeled 5,6-free two-graphs with n nodes.

1, 1, 2, 8, 52, 457, 4979, 64591, 972906, 16701834, 322063458, 6894918021, 162316253829, 4168330738093, 115980086558470, 3476156853885992, 111665862911781864, 3827642575341002133, 139457935266705019299, 5382149182666970080019, 219344947692643001216702

Offset: 1

Author

Peter J. Cameron

Keywords

• nonn

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 1..398
• P. J. Cameron, Counting two-graphs related to trees, Elec. J. Combin., Vol. 2, #R4.

Crossrefs

Cf. A007831.

Programs

Mathematica

nmax = 20; Rest[CoefficientList[Series[(3 + E^(2*x) - E^x*(4 + ProductLog[-1 + E^(-x)] * (2 + ProductLog[-1 + E^(-x)])))/(2*(-1 + E^x)), {x, 0, nmax}], x] * Range[0, nmax]!] (* Vaclav Kotesovec, Nov 16 2021 *)

PARI

seq(n)=my(e=exp(x + O(x^2*x^n))); Vec(serlaplace((3 + e^2 - e*(4 + lambertw(1/e - 1) * (2 + lambertw(1/e - 1)))) / (2*(e - 1)))) \\ Andrew Howroyd, Oct 15 2024

Formula

STIRLING transform of A007831. - Sean A. Irvine, Feb 03 2018

From Vaclav Kotesovec, Nov 16 2021: (Start)

E.g.f.: (3 + exp(2*x) - exp(x)*(4 + LambertW(exp(-x) - 1) * (2 + LambertW(exp(-x) - 1)))) / (2*(exp(x) - 1)).

a(n) ~ (exp(1) - 1)^(3/2) * (-log(1-exp(-1)))^(3/2) * n^(n-2) / (exp(n-1) * (1 - log(exp(1) - 1))^n). (End)