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A054914 Number of labeled connected digraphs with n nodes such that complement is also connected.

%I A054914 #23 Apr 29 2023 08:10:34
%S A054914 1,2,44,3572,1005584,1060875152,4382913876704,71987098738435232,
%T A054914 4721068803628864289024,1237845578934919489219757312,
%U A054914 1298046978912816702510086132201984,5444486716626952189940499391640815580672,91343710775311761525117954724021374685703481344
%N A054914 Number of labeled connected digraphs with n nodes such that complement is also connected.
%H A054914 Alois P. Heinz, <a href="/A054914/b054914.txt">Table of n, a(n) for n = 1..50</a>
%H A054914 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 A054914 a(n) = 2*A003027(n) - A053763(n).
%p A054914 b:= n-> 2^(n^2-n):
%p A054914 g:= proc(n) option remember; local k; `if`(n=0, 1,
%p A054914       b(n)- add(k*binomial(n,k) *b(n-k)*g(k), k=1..n-1)/n)
%p A054914     end:
%p A054914 a:= n-> 2*g(n)-b(n):
%p A054914 seq (a(n), n=1..20);  # _Alois P. Heinz_, Oct 21 2012
%t A054914 nn=20; g=Sum[2^(2Binomial[n,2])x^n/n!,{n,0,nn}];
%t A054914 Drop[Range[0,nn]!CoefficientList[Series[2(Log[g]+1)-g,{x,0,nn}],x],1]  (* _Geoffrey Critzer_, Oct 21 2012 *)
%o A054914 (Magma)
%o A054914 m:=30;
%o A054914 f:= func< x | (&+[2^(n*(n-1))*x^n/Factorial(n): n in [0..m+3]]) >;
%o A054914 R<x>:=PowerSeriesRing(Rationals(), m);
%o A054914 Coefficients(R!(Laplace( 1 + 2*Log(f(x)) - f(x) ))); // _G. C. Greubel_, Apr 28 2023
%o A054914 (SageMath)
%o A054914 m=30
%o A054914 def f(x): return sum(2^(n*(n-1))*x^n/factorial(n) for n in range(m+4))
%o A054914 def A054914_list(prec):
%o A054914     P.<x> = PowerSeriesRing(QQ, prec)
%o A054914     return P( 2 + 2*log(f(x)) - f(x) ).egf_to_ogf().list()
%o A054914 a=A054914_list(40); a[1:] # _G. C. Greubel_, Apr 28 2023
%Y A054914 Cf. A003027, A053763.
%K A054914 nonn,easy
%O A054914 1,2
%A A054914 _N. J. A. Sloane_, May 23 2000
%E A054914 More terms from _Vladeta Jovovic_, Jul 17 2000