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A054942 Number of connected oriented graphs on n nodes with an even number of edges.

%I A054942 #23 Apr 29 2023 08:10:39
%S A054942 1,0,12,304,27664,6990848,5179182272,11396324423680,74944172893348096,
%T A054942 1476405354971703541760,87208352627656970763963392,
%U A054942 15450530398306943408624578330624,8211400756816955708062672329408385024
%N A054942 Number of connected oriented graphs on n nodes with an even number of edges.
%H A054942 Andrew Howroyd, <a href="/A054942/b054942.txt">Table of n, a(n) for n = 1..50</a>
%H A054942 V. A. Liskovets, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL3/LISK/Derseq.html">Some easily derivable sequences</a>, J. Integer Sequences, 3 (2000), #00.2.2.
%F A054942 a(n) = (A054941(n) - (-1)^n*A000831(n-1))/2. - _Andrew Howroyd_, Sep 10 2018
%F A054942 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
%t A054942 nn = 15; g[z] := Sum[(1 + 2 u)^Binomial[n, 2] z^n/n!, {n, 0, nn}]; Drop[
%t A054942 Map[Total[#[[1 ;; Binomial[nn, 2] + 1 ;;2]]]&,Range[0,nn]!CoefficientList[
%t A054942 Series[Log[g[z]], {z, 0, nn}], {z, u}]], 1] (* _Geoffrey Critzer_, Jul 28 2016 *)
%o A054942 (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
%o A054942 (Magma)
%o A054942 m:=30;
%o A054942 f:= func< x | (&+[3^Binomial(n,2)*x^n/Factorial(n) : n in [0..m+3]]) >;
%o A054942 R<x>:=PowerSeriesRing(Rationals(), m);
%o A054942 Coefficients(R!(Laplace( Log(f(x)*(Cos(x) + Sin(x)))/2 ))); // _G. C. Greubel_, Apr 29 2023
%o A054942 (SageMath)
%o A054942 m=30
%o A054942 def f(x): return sum(3^binomial(n,2)*x^n/factorial(n) for n in range(m+4))
%o A054942 def A054941_list(prec):
%o A054942     P.<x> = PowerSeriesRing(QQ, prec)
%o A054942     return P( log(f(x)*(cos(x) + sin(x)))/2 ).egf_to_ogf().list()
%o A054942 a=A054941_list(40); a[1:] # _G. C. Greubel_, Apr 29 2023
%Y A054942 Cf. A000831, A054941, A054943.
%K A054942 nonn,easy
%O A054942 1,3
%A A054942 _N. J. A. Sloane_, May 24 2000
%E A054942 More terms from _Vladeta Jovovic_, Mar 11 2003