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

1, 2, 44, 3572, 1005584, 1060875152, 4382913876704, 71987098738435232, 4721068803628864289024, 1237845578934919489219757312, 1298046978912816702510086132201984, 5444486716626952189940499391640815580672, 91343710775311761525117954724021374685703481344

Offset: 1

Author

N. J. A. Sloane, May 23 2000

Keywords

• nonn
• easy

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.

Crossrefs

Cf. A003027, A053763.

Programs

Magma

m:=30;
f:= func< x | (&+[2^(n*(n-1))*x^n/Factorial(n): n in [0..m+3]]) >;
R:=PowerSeriesRing(Rationals(), m);
Coefficients(R!(Laplace( 1 + 2*Log(f(x)) - f(x) ))); // G. C. Greubel, Apr 28 2023

Maple

b:= n-> 2^(n^2-n):
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^(2Binomial[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 *)

SageMath

m=30
def f(x): return sum(2^(n*(n-1))*x^n/factorial(n) for n in range(m+4))
def A054914_list(prec):
    P. = PowerSeriesRing(QQ, prec)
    return P( 2 + 2*log(f(x)) - f(x) ).egf_to_ogf().list()
a=A054914_list(40); a[1:] # G. C. Greubel, Apr 28 2023

Formula

a(n) = 2*A003027(n) - A053763(n).

Extensions

More terms from Vladeta Jovovic, Jul 17 2000