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A277073 Number of n-node labeled graphs with two endpoints.

%I A277073 #26 Jan 12 2024 20:23:33
%S A277073 0,1,6,30,260,5445,228564,17288852,2327095296,562985438805,
%T A277073 248555982382840,203515251722217402,313711170518065772088,
%U A277073 922107609498513821505577,5221584862895700871908309960,57411615463478726571189869693160,1232855219250913685154581533108294112
%N A277073 Number of n-node labeled graphs with two endpoints.
%D A277073 F. Harary and E. Palmer, Graphical Enumeration, (1973), p. 31, problem 1.16(a).
%H A277073 Andrew Howroyd, <a href="/A277073/b277073.txt">Table of n, a(n) for n = 1..50</a>
%H A277073 Marko R. Riedel, Geoffrey Critzer, Math.Stackexchange.com, <a href="http://math.stackexchange.com/questions/1930410/">Proof of the closed form of the e.g.f. by combinatorial species</a>.
%F A277073 E.g.f.: (1/2)*(z^2/(1-z))*A(z) + (1/2)*(z^4/(1-z)^2)*(A''(z)-2*A'(z)+A(z)) + (1/2)*(z^3/(1-z)^3)*(A'(z)-A(z)) where A(z) = exp(1/2*z^2)*Sum_{n>=0} (2^binomial(n, 2)*(z/exp(z))^n/n!).
%p A277073 MX := 16:
%p A277073 XGF := exp(z^2/2)*add((z/exp(z))^n*2^binomial(n,2)/n!, n=0..MX+5):
%p A277073 K1 := 1/2*z^2/(1-z)*XGF:
%p A277073 K2 := 1/2*z^4/(1-z)^2*(diff(XGF, z$2)-2*diff(XGF,z)+XGF):
%p A277073 K3 := 1/2*z^3/(1-z)^3*(diff(XGF, z)-XGF):
%p A277073 XS := series(K1+K2+K3, z=0, MX+1):
%p A277073 seq(n!*coeff(XS, z, n), n=1..MX);
%t A277073 m = 16;
%t A277073 A[z_] := Exp[1/2*z^2]*Sum[2^Binomial[n, 2]*(z/Exp[z])^n/n!, {n, 0, m}];
%t A277073 egf = (1/2)*(z^2/(1 - z))*A[z] + (1/2)*(z^4/(1 - z)^2)*(A''[z] - 2*A'[z] + A[z]) + (1/2)*(z^3/(1 - z)^3)*(A'[z] - A[z]);
%t A277073 a[n_] := SeriesCoefficient[egf, {z, 0, n}]*n!;
%t A277073 Array[a, m] (* _Jean-François Alcover_, Feb 23 2019 *)
%Y A277073 Column k=2 of A327369.
%Y A277073 Cf. A059167, A277072, A277074.
%K A277073 nonn
%O A277073 1,3
%A A277073 _Marko Riedel_, Sep 27 2016