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%I #17 Oct 15 2024 18:07:08
%S 1,1,2,8,52,457,4979,64591,972906,16701834,322063458,6894918021,
%T 162316253829,4168330738093,115980086558470,3476156853885992,
%U 111665862911781864,3827642575341002133,139457935266705019299,5382149182666970080019,219344947692643001216702
%N Number of point labeled 5,6-free two-graphs with n nodes.
%H Vaclav Kotesovec, <a href="/A007832/b007832.txt">Table of n, a(n) for n = 1..398</a>
%H 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 STIRLING transform of A007831. - _Sean A. Irvine_, Feb 03 2018
%F From _Vaclav Kotesovec_, Nov 16 2021: (Start)
%F E.g.f.: (3 + exp(2*x) - exp(x)*(4 + LambertW(exp(-x) - 1) * (2 + LambertW(exp(-x) - 1)))) / (2*(exp(x) - 1)).
%F 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 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 (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 Cf. A007831.
%K nonn
%O 1,3
%A _Peter J. Cameron_