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A277072
Number of n-node labeled graphs with one endpoint.
5
0, 0, 0, 12, 320, 10890, 640836, 68362504, 13369203792, 4852623272670, 3314874720579180, 4318786169776866612, 10854838945689940034808, 53111101422881446287824390, 509319855642185873306564196780, 9619620856997967197817249800046480
OFFSET
1,4
REFERENCES
F. Harary and E. Palmer, Graphical Enumeration, (1973), p. 31, problem 1.16(a).
LINKS
Marko R. Riedel, Geoffrey Critzer, Math.Stackexchange.com, Proof of the closed form of the e.g.f. by combinatorial species.
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!).
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 *)
CROSSREFS
Column k=1 of A327369.
Sequence in context: A341185 A279293 A180790 * A080325 A083431 A239781
KEYWORD
nonn
AUTHOR
Marko Riedel, Sep 27 2016
STATUS
approved