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A277073
Number of n-node labeled graphs with two endpoints.
5
0, 1, 6, 30, 260, 5445, 228564, 17288852, 2327095296, 562985438805, 248555982382840, 203515251722217402, 313711170518065772088, 922107609498513821505577, 5221584862895700871908309960, 57411615463478726571189869693160, 1232855219250913685154581533108294112
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
Column k=2 of A327369.
Sequence in context: A133668 A121772 A270845 * A052585 A304188 A343574
KEYWORD
nonn
AUTHOR
Marko Riedel, Sep 27 2016
STATUS
approved