A065456 Number of functions on n labeled nodes whose representation as a digraph has two components.

0, 1, 9, 95, 1220, 18694, 334369, 6852460, 158479488, 4085349936, 116193701393, 3615197586912, 122165572502324, 4456126288810624, 174520484866919385, 7304657490838627072, 325420940777809245152, 15374940186972235659264, 767898500931828204443769

Offset: 1

Author

John W. Layman, Nov 24 2001

Keywords

• nonn

Examples

			a(3)=9 since, on {1,2,3}, these functions and no others have two components: (3->1->3)(2->2), (1->3->1)(2->2), (3->2->2)(1->1), (2->3->2)(1->1), (2->1->2)(3->3), (1->2->1)(3->3), (1->2->2)(3->3), (1->3->3)(2->2) and (2->3->3)(1->1).
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..386

Crossrefs

Column k=2 of A060281.

See A001865 for the numbers of one-component (i.e. connected) functions on n labeled nodes.

Programs

Maple

katz := n->(n-1)!*sum(n^k/k!,k=0..n-1); A001865 := []; for m from 1 to 30 do A001865 := [op(A001865),katz(m)] od; A065456 := []; for n from 1 to 29 do unequal_splits := sum(binomial(n,k)*A001865[k]*A001865[n-k],k=1..floor((n-1)/2)); if (n mod 2=0) then A065456 := [op(A065456),unequal_splits+binomial(n,n/2)*(A001865[n/2])^2/2] fi; if (n mod 2=1) then A065456 := [op(A065456),unequal_splits] fi od; print(A065456); #if the connected components are of equal size, we correct the double counting. The Katz reference is at A001865. - Len Smiley, Nov 26 2001
# second Maple program:
g:= proc(n) option remember; add(n^(n-j)*(n-1)!/(n-j)!, j=1..n) end:
a:= n-> add(binomial(n, i)*g(i)*g(n-i)/2, i=0..n):
seq(a(n), n=1..20);  # Alois P. Heinz, Dec 19 2021

Mathematica

t=Sum[n^(n-1)x^n/n!,{n,1,20}];  Range[0, 20]! CoefficientList[Series[Log[1/(1 - t)]^2/2, {x, 0, 20}],
x] (* Geoffrey Critzer, Oct 06 2011 *)
Rest[CoefficientList[Series[Log[1+LambertW[-x]]^2, {x, 0, 20}], x]/2* Range[0, 20]!] (* Vaclav Kotesovec, Oct 05 2013 *)

PARI

x='x+O('x^20); concat([0], Vec(serlaplace(log(1+lambertw(-x))^2/2 ))) \\ G. C. Greubel, Jan 18 2018

Formula

E.g.f.: 1/2 * log(1+LambertW(-x))^2. - Vladeta Jovovic, Nov 25 2001

a(n) ~ (n-1)! * exp(n)*(log(n/2) + gamma)/4, where gamma is the Euler-Mascheroni constant (A001620). - Vaclav Kotesovec, Oct 05 2013

Extensions

More terms from Vladeta Jovovic, Nov 25 2001