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
KEYWORD
nonn
AUTHOR
Marko Riedel, Sep 27 2016
STATUS
approved