A277074 Number of n-node labeled graphs with three endpoints.

0, 0, 0, 4, 80, 1860, 64680, 3666600, 354093264, 59372032440, 17572209206640, 9347625940951980, 9099961952914672840, 16480899322963497105684, 56311549004017312945310280, 367105988116570172056739960080

Offset

1,4

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.

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-François Alcover, Feb 23 2019 *)

Crossrefs

Column k=3 of A327369.

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