A054913 Number of labeled connected graphs with n nodes such that complement is also connected.

1, 0, 0, 12, 432, 20640, 1635360, 234661728, 63873105408, 33808605100800, 35254518078942720, 72922216118695037952, 300312950395670884227072, 2467417543490126920783534080, 40490542668157619621325008117760, 1327929920886650529112870913410510848

Offset

1,4

Links

Alois P. Heinz, Table of n, a(n) for n = 1..50

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

Cf. A001187, A006125.

Sequence in context: A347795 A129006 A067429 * A221955 A070285 A241593

Adjacent sequences: A054910 A054911 A054912 * A054914 A054915 A054916

Keyword

nonn,easy

Author

N. J. A. Sloane, May 23 2000

Extensions

More terms from Vladeta Jovovic, Jul 19 2000

Status

approved