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A277074 Number of n-node labeled graphs with three endpoints. 3
0, 0, 0, 4, 80, 1860, 64680, 3666600, 354093264, 59372032440, 17572209206640, 9347625940951980, 9099961952914672840, 16480899322963497105684, 56311549004017312945310280, 367105988116570172056739960080 (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.: (1/6)*(z^4/(1-z)^3)*A(z) + (1/2)*(z^4/(1-z)^2)*(A'(z)-A(z)) + (1/6)*(z^6/(1-z)^3)*(A'''(z)-3*A''(z)+3*A'(z)-A(z)) + (1/2)*(z^5/(1-z)^4)*(A''(z)-2*A'(z)+A(z)) + (1/6)*(z^4/(1-z)^4)*(A'(z)-A(z)) + (1/2)*(z^5/(1-z)^5)*(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 := 1/6*z^4/(1-z)^3*XGF:

K2 := 1/2*z^4/(1-z)^2*(diff(XGF, z)-XGF):

K3 := 1/6*z^6/(1-z)^3*(diff(XGF, z$3)-3*diff(XGF, z$2)+3*diff(XGF, z)-XGF):

K4 := 1/2*z^5/(1-z)^4*(diff(XGF, z$2)-2*diff(XGF, z)+XGF):

K5 := 1/6*z^4/(1-z)^4*(diff(XGF, z)-XGF):

K6 := 1/2*z^5/(1-z)^5*(diff(XGF, z)-XGF):

XS := series(K1+K2+K3+K4+K5+K6, 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+1}];

egf = (1/6)*(z^4/(1-z)^3)*A[z] + (1/2)*(z^4/(1-z)^2)*(A'[z] - A[z]) + (1/6)*(z^6/(1-z)^3)*(A'''[z] - 3*A''[z] + 3*A'[z] - A[z]) + (1/2)*(z^5/(1 - z)^4)*(A''[z] - 2*A'[z] + A[z]) + (1/6)*(z^4/(1-z)^4)*(A'[z] - A[z]) + (1/2)*(z^5/(1-z)^5)*(A'[z] - A[z]); s = egf + O[z]^(m+1);

a[n_] := n!*SeriesCoefficient[s, n];

Array[a, m] (* Jean-Fran├žois Alcover, Feb 23 2019 *)

CROSSREFS

Cf. A059167, A277072, A277073.

Sequence in context: A114488 A055787 A132584 * A012127 A189791 A057875

Adjacent sequences:  A277071 A277072 A277073 * A277075 A277076 A277077

KEYWORD

nonn

AUTHOR

Marko Riedel, Sep 27 2016

STATUS

approved

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Last modified November 27 07:28 EST 2021. Contains 349365 sequences. (Running on oeis4.)