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 A277073 Number of n-node labeled graphs with two endpoints. 4
 0, 1, 6, 30, 260, 5445, 228564, 17288852, 2327095296, 562985438805, 248555982382840, 203515251722217402, 313711170518065772088, 922107609498513821505577, 5221584862895700871908309960, 57411615463478726571189869693160, 1232855219250913685154581533108294112 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,3 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.: (1/2)*(z^2/(1-z))*A(z) + (1/2)*(z^4/(1-z)^2)*(A''(z)-2*A'(z)+A(z)) + (1/2)*(z^3/(1-z)^3)*(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 := 1/2*z^2/(1-z)*XGF: K2 := 1/2*z^4/(1-z)^2*(diff(XGF, z\$2)-2*diff(XGF, z)+XGF): K3 := 1/2*z^3/(1-z)^3*(diff(XGF, z)-XGF): XS := series(K1+K2+K3, 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 = (1/2)*(z^2/(1 - z))*A[z] + (1/2)*(z^4/(1 - z)^2)*(A''[z] - 2*A'[z] + A[z]) + (1/2)*(z^3/(1 - z)^3)*(A'[z] - A[z]); a[n_] := SeriesCoefficient[egf, {z, 0, n}]*n!; Array[a, m] (* Jean-François Alcover, Feb 23 2019 *) CROSSREFS Cf. A059167, A277072, A277074. Sequence in context: A133668 A121772 A270845 * A052585 A304188 A343574 Adjacent sequences:  A277070 A277071 A277072 * A277074 A277075 A277076 KEYWORD nonn AUTHOR Marko Riedel, Sep 27 2016 STATUS approved

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Last modified June 23 08:51 EDT 2021. Contains 345395 sequences. (Running on oeis4.)