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

0, 1, 6, 30, 260, 5445, 228564, 17288852, 2327095296, 562985438805, 248555982382840, 203515251722217402, 313711170518065772088, 922107609498513821505577, 5221584862895700871908309960, 57411615463478726571189869693160, 1232855219250913685154581533108294112

Offset: 1

Author

Marko Riedel, Sep 27 2016

Keywords

• nonn

References

• F. Harary and E. Palmer, Graphical Enumeration, (1973), p. 31, problem 1.16(a).

Links

• Andrew Howroyd, Table of n, a(n) for n = 1..50
• Marko R. Riedel, Geoffrey Critzer, Math.Stackexchange.com, Proof of the closed form of the e.g.f. by combinatorial species.

Crossrefs

Column k=2 of A327369.

Cf. A059167, A277072, A277074.

Programs

Maple

MX := 16:
XGF := exp(z^2/2)*add((z/exp(z))^n*2^binomial(n,2)/n!, n=0..MX+5):
K1 := 1/2*z^2/(1-z)*XGF:
K2 := 1/2*z^4/(1-z)^2*(diff(XGF, z$2)-2*diff(XGF,z)+XGF):
K3 := 1/2*z^3/(1-z)^3*(diff(XGF, z)-XGF):
XS := series(K1+K2+K3, z=0, MX+1):
seq(n!*coeff(XS, z, n), n=1..MX);

Mathematica

m = 16;
A[z_] := Exp[1/2*z^2]*Sum[2^Binomial[n, 2]*(z/Exp[z])^n/n!, {n, 0, m}];
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]);
a[n_] := SeriesCoefficient[egf, {z, 0, n}]*n!;
Array[a, m] (* Jean-François Alcover, Feb 23 2019 *)

Formula

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!).