cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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

This page as a plain text file.
%I A007832 #17 Oct 15 2024 18:07:08
%S A007832 1,1,2,8,52,457,4979,64591,972906,16701834,322063458,6894918021,
%T A007832 162316253829,4168330738093,115980086558470,3476156853885992,
%U A007832 111665862911781864,3827642575341002133,139457935266705019299,5382149182666970080019,219344947692643001216702
%N A007832 Number of point labeled 5,6-free two-graphs with n nodes.
%H A007832 Vaclav Kotesovec, <a href="/A007832/b007832.txt">Table of n, a(n) for n = 1..398</a>
%H A007832 P. J. Cameron, <a href="http://www.combinatorics.org/Volume_2/volume2.html#R4">Counting two-graphs related to trees</a>, Elec. J. Combin., Vol. 2, #R4.
%F A007832 STIRLING transform of A007831. - _Sean A. Irvine_, Feb 03 2018
%F A007832 From _Vaclav Kotesovec_, Nov 16 2021: (Start)
%F A007832 E.g.f.: (3 + exp(2*x) - exp(x)*(4 + LambertW(exp(-x) - 1) * (2 + LambertW(exp(-x) - 1)))) / (2*(exp(x) - 1)).
%F A007832 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)
%t A007832 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 *)
%o A007832 (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
%Y A007832 Cf. A007831.
%K A007832 nonn
%O A007832 1,3
%A A007832 _Peter J. Cameron_