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A277072 Number of n-node labeled graphs with one endpoint. 4
0, 0, 0, 12, 320, 10890, 640836, 68362504, 13369203792, 4852623272670, 3314874720579180, 4318786169776866612, 10854838945689940034808, 53111101422881446287824390, 509319855642185873306564196780, 9619620856997967197817249800046480 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,4

REFERENCES

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

LINKS

Table of n, a(n) for n=1..16.

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-Fran├žois Alcover, Feb 23 2019 *)

CROSSREFS

Cf. A059167, A277073, A277074.

Sequence in context: A341185 A279293 A180790 * A080325 A083431 A239781

Adjacent sequences:  A277069 A277070 A277071 * A277073 A277074 A277075

KEYWORD

nonn

AUTHOR

Marko Riedel, Sep 27 2016

STATUS

approved

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Last modified May 19 12:20 EDT 2022. Contains 353833 sequences. (Running on oeis4.)