A277072 - OEIS

%I #23 Jan 12 2024 20:26:34

%S 0,0,0,12,320,10890,640836,68362504,13369203792,4852623272670,

%T 3314874720579180,4318786169776866612,10854838945689940034808,

%U 53111101422881446287824390,509319855642185873306564196780,9619620856997967197817249800046480

%N Number of n-node labeled graphs with one endpoint.

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

%H Andrew Howroyd, <a href="/A277072/b277072.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.: (z^2/(1-z))*(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 := z^2/(1-z)*(diff(XGF,z)-XGF):

%p XS := series(K1, 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}];

%t egf = (z^2/(1 - z))*(A'[z] - A[z]);

%t a[n_] := SeriesCoefficient[egf, {z, 0, n}]*n!;

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

%Y Column k=1 of A327369.

%Y Cf. A059167, A277073, A277074.

%K nonn

%O 1,4

%A _Marko Riedel_, Sep 27 2016