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%I #22 Sep 08 2022 08:44:59
%S 0,0,0,0,6,40,260,1848,14616,128448,1246752,13273920,153996480,
%T 1935048960,26193473280,380120670720,5888620684800,97007636275200,
%U 1693590745190400,31237853849395200,607035345406156800
%N Number of pairs of cycles of cardinality at least 2.
%H G. C. Greubel, <a href="/A052518/b052518.txt">Table of n, a(n) for n = 0..445</a>
%H INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=84">Encyclopedia of Combinatorial Structures 84</a>
%F E.g.f.: log(1-x)^2 + 2*x*log(1-x) + x^2.
%F n*a(n+2) + (1-n-2*n^2)*a(n+1) - n*(1-n^2)*a(n) = 0, with a(0) = ... = a(3) = 0, a(4) = 3!.
%F a(n) = 2*(n-2)!*((n-1)*(Psi(n) + gamma) - n), n>2. - _Vladeta Jovovic_, Sep 21 2003
%p Pairs spec := [S,{B=Cycle(Z,2 <= card),S=Prod(B,B)},labeled]: seq(combstruct[count](spec,size=n), n=0..20);
%t With[{m = 25}, CoefficientList[Series[Log[1-x]^2 +2*x*Log[1-x] +x^2, {x, 0, m}], x]*Range[0, m]!] (* _G. C. Greubel_, May 13 2019 *)
%o (PARI) a(n) = if (n <= 2, 0, round(2*(n-2)!*((n-1)*(psi(n)+Euler)-n))); \\ _Michel Marcus_, Jul 08 2015
%o (PARI) my(x='x+O('x^25)); concat(vector(4), Vec(serlaplace( log(1-x)^2 + 2*x*log(1-x) + x^2 ))) \\ _G. C. Greubel_, May 13 2019
%o (Magma) m:=25; R<x>:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!( Log(1-x)^2 + 2*x*Log(1-x) + x^2 )); [0,0,0,0] cat [Factorial(n+3)*b[n]: n in [1..m-4]]; // _G. C. Greubel_, May 13 2019
%o (Sage) m = 25; T = taylor(log(1-x)^2 + 2*x*log(1-x) + x^2, x, 0, m); [factorial(n)*T.coefficient(x, n) for n in (0..m)] # _G. C. Greubel_, May 13 2019
%Y Cf. A000254, A000276.
%K easy,nonn
%O 0,5
%A encyclopedia(AT)pommard.inria.fr, Jan 25 2000
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