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A054913
Number of labeled connected graphs with n nodes such that complement is also connected.
1
1, 0, 0, 12, 432, 20640, 1635360, 234661728, 63873105408, 33808605100800, 35254518078942720, 72922216118695037952, 300312950395670884227072, 2467417543490126920783534080, 40490542668157619621325008117760, 1327929920886650529112870913410510848
OFFSET
1,4
LINKS
V. A. Liskovets, Some easily derivable sequences, J. Integer Sequences, 3 (2000), #00.2.2.
FORMULA
a(n) = 2*A001187(n) - A006125(n).
MAPLE
b:= n-> 2^(n*(n-1)/2):
g:= proc(n) option remember; local k; `if`(n=0, 1,
b(n)- add(k*binomial(n, k) *b(n-k)*g(k), k=1..n-1)/n)
end:
a:= n-> 2*g(n)-b(n):
seq (a(n), n=1..20); # Alois P. Heinz, Oct 21 2012
MATHEMATICA
nn=20; g=Sum[2^Binomial[n, 2]x^n/n!, {n, 0, nn}];
Drop[Range[0, nn]!CoefficientList[Series[2(Log[g]+1)-g, {x, 0, nn}], x], 1] (* Geoffrey Critzer, Oct 21 2012 *)
PROG
(Magma)
m:=30;
f:= func< x | (&+[2^Binomial(n, 2)*x^n/Factorial(n) : n in [0..m+3]]) >;
R<x>:=PowerSeriesRing(Rationals(), m);
Coefficients(R!(Laplace( 1 + 2*Log(f(x)) - f(x) ))); // G. C. Greubel, Apr 28 2023
(SageMath)
m=30
def f(x): return sum(2^binomial(n, 2)*x^n/factorial(n) for n in range(m+4))
def A054913_list(prec):
P.<x> = PowerSeriesRing(QQ, prec)
return P( 2 +2*log(f(x)) -f(x) ).egf_to_ogf().list()
a=A054913_list(40); a[1:] # G. C. Greubel, Apr 28 2023
CROSSREFS
Sequence in context: A347795 A129006 A067429 * A221955 A070285 A241593
KEYWORD
nonn,easy
AUTHOR
N. J. A. Sloane, May 23 2000
EXTENSIONS
More terms from Vladeta Jovovic, Jul 19 2000
STATUS
approved