A054941 - OEIS

%I #31 Apr 29 2023 08:10:29

%S 1,2,20,624,55248,13982208,10358360640,22792648882176,

%T 149888345786341632,2952810709943411146752,

%U 174416705255313941476193280,30901060796613886817249881227264,16422801513633911416125344647746244608,26183660776604240464418800095675915958222848

%N Number of weakly connected oriented graphs on n labeled nodes.

%C The triangle of oriented labeled graphs on n>=1 nodes with 1<=k<=n components and row sums A047656 starts:

%C        1;

%C        2,      1;

%C       20,      6,     1;

%C      624,     92,    12,   1;

%C    55248,   3520,   260,  20,  1;

%C 13982208, 354208, 11880, 580, 30, 1; - _R. J. Mathar_, Apr 29 2019

%H Seiichi Manyama, <a href="/A054941/b054941.txt">Table of n, a(n) for n = 1..65</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 E.g.f.: log( Sum_{n >= 0} 3^binomial(n, 2)*x^n/n! ). - _Vladeta Jovovic_, Feb 14 2003

%t nn=20; s=Sum[3^Binomial[n,2]x^n/n!,{n,0,nn}];

%t Drop[Range[0,nn]! CoefficientList[Series[Log[s]+1,{x,0,nn}],x],1] (* _Geoffrey Critzer_, Oct 22 2012 *)

%o (PARI) N=20; x='x+O('x^N); Vec(serlaplace(log(sum(k=0, N, 3^binomial(k, 2)*x^k/k!)))) \\ _Seiichi Manyama_, May 18 2019

%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)) ))); // _G. C. Greubel_, Apr 28 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)) ).egf_to_ogf().list()

%o a=A054941_list(40); a[1:] # _G. C. Greubel_, Apr 28 2023

%Y Row sums of A350732.

%Y The unlabeled version is A086345.

%Y Cf. A001187 (graphs), A003027 (digraphs), A350730 (strongly connected).

%K nonn,easy

%O 1,2

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

%E More terms from _Vladeta Jovovic_, Feb 14 2003