A054948 Number of labeled semi-strong digraphs on n nodes.

1, 1, 2, 22, 1688, 573496, 738218192, 3528260038192, 63547436065854848, 4400982906402148836736, 1190477715153930158152163072, 1270476960212865235273469079407872, 5381083212549793228422395071641588743168, 90765858793484859057065439213726376149311958016

Offset

0,3

Comments

A digraph is semi-strong if all its weakly connected components are strongly connected. - Andrew Howroyd, Jan 14 2022

A digraph is semi-strong iff the following implication holds for all x,y in [n]: If there is a directed path from x to y then x and y are in the same strongly connected component. - Geoffrey Critzer, Oct 07 2023

Links

Andrew Howroyd, Table of n, a(n) for n = 0..50

V. A. Liskovets, Some easily derivable sequences, J. Integer Sequences, 3 (2000), #00.2.2.

Formula

E.g.f.: 1/(1-B(x)) where B(x) is e.g.f. for A054947. - Vladeta Jovovic, Mar 11 2003

E.g.f. A(x) satisfies: Sum_{n>=0} 2^(n^2-n)*x^n/n! / A(2^n*x) = 1. - Paul D. Hanna, Oct 27 2012

E.g.f.: exp(B(x)) where B(x) is the e.g.f. of A003030. - Andrew Howroyd, Jan 14 2022

Mathematica

a[n_] := a[n] = Module[{A}, A = 1+Sum[a[k]*x^k/k!, {k, 1, n-1}]; n!*SeriesCoefficient[Sum[2^(k^2-k)*x^k/k!/(A /. x -> 2^k*x) , {k, 0, n}], {x, 0, n}]]; a[0]=1; Table[a[n], {n, 0, 13}] (* Jean-François Alcover, Dec 15 2014, after Paul D. Hanna *)

Prog

(PARI) a(n)=local(A=1+sum(k=1, n-1, a(k)*x^k/k!)+x*O(x^n)); n!*polcoeff(sum(k=0, n, 2^(k^2-k)*x^k/k!/subst(A, x, 2^k*x)), n)
for(n=0, 10, print1(a(n), ", ")) \\ Paul D. Hanna, Oct 27 2012

Crossrefs

The unlabeled version is A350754.

Cf. A003030, A054947, A218393.

Sequence in context: A177410 A193486 A337577 * A330124 A104149 A113761

Adjacent sequences: A054945 A054946 A054947 * A054949 A054950 A054951

Keyword

nonn,easy

Author

N. J. A. Sloane, May 24 2000

Extensions

More terms from Vladeta Jovovic, Mar 11 2003

Changed offset to 0 and added a(0)=1 by Paul D. Hanna, Oct 27 2012

Status

approved