A277072 Number of n-node labeled graphs with one endpoint.

0, 0, 0, 12, 320, 10890, 640836, 68362504, 13369203792, 4852623272670, 3314874720579180, 4318786169776866612, 10854838945689940034808, 53111101422881446287824390, 509319855642185873306564196780, 9619620856997967197817249800046480

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=1 of A327369.

Cf. A059167, A277073, 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 := z^2/(1-z)*(diff(XGF,z)-XGF):
XS := series(K1, 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 = (z^2/(1 - z))*(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.: (z^2/(1-z))*(A'(z)-A(z)) where A(z) = exp(1/2*z^2) * Sum_{n>=0}(2^binomial(n, 2)*(z/exp(z))^n/n!).