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A054942 Number of connected oriented graphs on n nodes with an even number of edges. 2

%I #23 Apr 29 2023 08:10:39

%S 1,0,12,304,27664,6990848,5179182272,11396324423680,74944172893348096,

%T 1476405354971703541760,87208352627656970763963392,

%U 15450530398306943408624578330624,8211400756816955708062672329408385024

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

%H Andrew Howroyd, <a href="/A054942/b054942.txt">Table of n, a(n) for n = 1..50</a>

%H 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 a(n) = (A054941(n) - (-1)^n*A000831(n-1))/2. - _Andrew Howroyd_, Sep 10 2018

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

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

%t Series[Log[g[z]], {z, 0, nn}], {z, u}]], 1] (* _Geoffrey Critzer_, Jul 28 2016 *)

%o (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 (Magma)

%o m:=30;

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

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

%o Coefficients(R!(Laplace( Log(f(x)*(Cos(x) + Sin(x)))/2 ))); // _G. C. Greubel_, Apr 29 2023

%o (SageMath)

%o m=30

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

%o def A054941_list(prec):

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

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

%o a=A054941_list(40); a[1:] # _G. C. Greubel_, Apr 29 2023

%Y Cf. A000831, A054941, A054943.

%K nonn,easy

%O 1,3

%A _N. J. A. Sloane_, May 24 2000

%E More terms from _Vladeta Jovovic_, Mar 11 2003

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Last modified March 19 01:57 EDT 2024. Contains 370952 sequences. (Running on oeis4.)