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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 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 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.)