login
Number of n-node labeled graphs with three endpoints.
4

%I #22 Jan 12 2024 20:27:28

%S 0,0,0,4,80,1860,64680,3666600,354093264,59372032440,17572209206640,

%T 9347625940951980,9099961952914672840,16480899322963497105684,

%U 56311549004017312945310280,367105988116570172056739960080

%N Number of n-node labeled graphs with three endpoints.

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

%H Andrew Howroyd, <a href="/A277074/b277074.txt">Table of n, a(n) for n = 1..50</a>

%H Marko R. Riedel, Geoffrey Critzer, Math.Stackexchange.com, <a href="http://math.stackexchange.com/questions/1930410/">Proof of the closed form of the e.g.f. by combinatorial species</a>.

%F 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!.

%p MX := 16:

%p XGF := exp(z^2/2)*add((z/exp(z))^n*2^binomial(n,2)/n!, n=0..MX+5):

%p K1 := 1/6*z^4/(1-z)^3*XGF:

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

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

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

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

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

%p XS := series(K1+K2+K3+K4+K5+K6, z=0, MX+1):

%p seq(n!*coeff(XS, z, n), n=1..MX);

%t m = 16;

%t A[z_] := Exp[1/2*z^2]*Sum[2^Binomial[n, 2]*(z/Exp[z])^n/n!, {n, 0, m+1}];

%t 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);

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

%t Array[a, m] (* _Jean-François Alcover_, Feb 23 2019 *)

%Y Column k=3 of A327369.

%Y Cf. A059167, A277072, A277073.

%K nonn

%O 1,4

%A _Marko Riedel_, Sep 27 2016