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%I #34 Dec 19 2021 13:24:20
%S 0,1,9,95,1220,18694,334369,6852460,158479488,4085349936,116193701393,
%T 3615197586912,122165572502324,4456126288810624,174520484866919385,
%U 7304657490838627072,325420940777809245152,15374940186972235659264,767898500931828204443769
%N Number of functions on n labeled nodes whose representation as a digraph has two components.
%H Alois P. Heinz, <a href="/A065456/b065456.txt">Table of n, a(n) for n = 1..386</a>
%F E.g.f.: 1/2 * log(1+LambertW(-x))^2. - _Vladeta Jovovic_, Nov 25 2001
%F a(n) ~ (n-1)! * exp(n)*(log(n/2) + gamma)/4, where gamma is the Euler-Mascheroni constant (A001620). - _Vaclav Kotesovec_, Oct 05 2013
%e 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).
%p 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
%p # second Maple program:
%p g:= proc(n) option remember; add(n^(n-j)*(n-1)!/(n-j)!, j=1..n) end:
%p a:= n-> add(binomial(n, i)*g(i)*g(n-i)/2, i=0..n):
%p seq(a(n), n=1..20); # _Alois P. Heinz_, Dec 19 2021
%t 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}],
%t x] (* _Geoffrey Critzer_, Oct 06 2011 *)
%t Rest[CoefficientList[Series[Log[1+LambertW[-x]]^2, {x, 0, 20}], x]/2* Range[0, 20]!] (* _Vaclav Kotesovec_, Oct 05 2013 *)
%o (PARI) x='x+O('x^20); concat([0], Vec(serlaplace(log(1+lambertw(-x))^2/2 ))) \\ _G. C. Greubel_, Jan 18 2018
%Y Column k=2 of A060281.
%Y See A001865 for the numbers of one-component (i.e. connected) functions on n labeled nodes.
%K nonn
%O 1,3
%A _John W. Layman_, Nov 24 2001
%E More terms from _Vladeta Jovovic_, Nov 25 2001
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