A054913 - OEIS

%I #25 Apr 29 2023 08:10:43

%S 1,0,0,12,432,20640,1635360,234661728,63873105408,33808605100800,

%T 35254518078942720,72922216118695037952,300312950395670884227072,

%U 2467417543490126920783534080,40490542668157619621325008117760,1327929920886650529112870913410510848

%N Number of labeled connected graphs with n nodes such that complement is also connected.

%H Alois P. Heinz, <a href="/A054913/b054913.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) = 2*A001187(n) - A006125(n).

%p b:= n-> 2^(n*(n-1)/2):

%p g:= proc(n) option remember; local k; `if`(n=0, 1,

%p       b(n)- add(k*binomial(n,k) *b(n-k)*g(k), k=1..n-1)/n)

%p     end:

%p a:= n-> 2*g(n)-b(n):

%p seq (a(n), n=1..20);  # _Alois P. Heinz_, Oct 21 2012

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

%t Drop[Range[0,nn]!CoefficientList[Series[2(Log[g]+1)-g, {x,0,nn}],x],1] (* _Geoffrey Critzer_, Oct 21 2012 *)

%o (Magma)

%o m:=30;

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

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

%o Coefficients(R!(Laplace( 1 + 2*Log(f(x)) - f(x) ))); // _G. C. Greubel_, Apr 28 2023

%o (SageMath)

%o m=30

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

%o def A054913_list(prec):

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

%o     return P( 2 +2*log(f(x)) -f(x) ).egf_to_ogf().list()

%o a=A054913_list(40); a[1:] # _G. C. Greubel_, Apr 28 2023

%Y Cf. A001187, A006125.

%K nonn,easy

%O 1,4

%A _N. J. A. Sloane_, May 23 2000

%E More terms from _Vladeta Jovovic_, Jul 19 2000