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A065456 Number of functions on n labeled nodes whose representation as a digraph has two components. 4
0, 1, 9, 95, 1220, 18694, 334369, 6852460, 158479488, 4085349936, 116193701393, 3615197586912, 122165572502324, 4456126288810624, 174520484866919385, 7304657490838627072, 325420940777809245152, 15374940186972235659264, 767898500931828204443769 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,3
LINKS
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
EXAMPLE
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).
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 *)
PROG
(PARI) x='x+O('x^20); concat([0], Vec(serlaplace(log(1+lambertw(-x))^2/2 ))) \\ G. C. Greubel, Jan 18 2018
CROSSREFS
Column k=2 of A060281.
See A001865 for the numbers of one-component (i.e. connected) functions on n labeled nodes.
Sequence in context: A213019 A098450 A337269 * A052389 A197665 A064504
KEYWORD
nonn
AUTHOR
John W. Layman, Nov 24 2001
EXTENSIONS
More terms from Vladeta Jovovic, Nov 25 2001
STATUS
approved

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Last modified April 13 02:05 EDT 2024. Contains 371639 sequences. (Running on oeis4.)