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A054942

Number of connected oriented graphs on n nodes with an even number of edges.

1, 0, 12, 304, 27664, 6990848, 5179182272, 11396324423680, 74944172893348096, 1476405354971703541760, 87208352627656970763963392, 15450530398306943408624578330624, 8211400756816955708062672329408385024

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,3

LINKS

Andrew Howroyd, Table of n, a(n) for n = 1..50

V. A. Liskovets, Some easily derivable sequences, J. Integer Sequences, 3 (2000), #00.2.2.

FORMULA

a(n) = (A054941(n) - (-1)^n*A000831(n-1))/2. - Andrew Howroyd, Sep 10 2018

E.g.f.: log(f(x)*(cos(x) + sin(x))), where f(x) = Sum_{j >= 0} 3^binomial(j, 2)*x^j/j!. - G. C. Greubel, Apr 29 2023

MATHEMATICA

nn = 15; g[z] := Sum[(1 + 2 u)^Binomial[n, 2] z^n/n!, {n, 0, nn}]; Drop[

Map[Total[#[[1 ;; Binomial[nn, 2] + 1 ;; 2]]]&, Range[0, nn]!CoefficientList[

Series[Log[g[z]], {z, 0, nn}], {z, u}]], 1] (* Geoffrey Critzer, Jul 28 2016 *)

PROG

(PARI) seq(n)={my(A=O(x*x^n)); Vec(serlaplace(log(sum(k=0, n, 3^binomial(k, 2)*x^k/k!) + A) + log(cos(x + A) + sin(x + A)))/2)} \\ Andrew Howroyd, Sep 10 2018

(Magma)

m:=30;

f:= func< x | (&+[3^Binomial(n, 2)*x^n/Factorial(n) : n in [0..m+3]]) >;

R<x>:=PowerSeriesRing(Rationals(), m);

Coefficients(R!(Laplace( Log(f(x)*(Cos(x) + Sin(x)))/2 ))); // G. C. Greubel, Apr 29 2023

(SageMath)

m=30

def f(x): return sum(3^binomial(n, 2)*x^n/factorial(n) for n in range(m+4))

def A054941_list(prec):

P.<x> = PowerSeriesRing(QQ, prec)

return P( log(f(x)*(cos(x) + sin(x)))/2 ).egf_to_ogf().list()

a=A054941_list(40); a[1:] # G. C. Greubel, Apr 29 2023