A052518 Number of pairs of cycles of cardinality at least 2.

0, 0, 0, 0, 6, 40, 260, 1848, 14616, 128448, 1246752, 13273920, 153996480, 1935048960, 26193473280, 380120670720, 5888620684800, 97007636275200, 1693590745190400, 31237853849395200, 607035345406156800

Offset: 0

Author

encyclopedia(AT)pommard.inria.fr, Jan 25 2000

Keywords

• easy
• nonn

Links

• G. C. Greubel, Table of n, a(n) for n = 0..445
• INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 84

Crossrefs

Cf. A000254, A000276.

Programs

Magma

m:=25; R:=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

Maple

Pairs spec := [S,{B=Cycle(Z,2 <= card),S=Prod(B,B)},labeled]: seq(combstruct[count](spec,size=n), n=0..20);

Mathematica

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 *)

PARI

a(n) = if (n <= 2, 0, round(2*(n-2)!*((n-1)*(psi(n)+Euler)-n))); \\ Michel Marcus, Jul 08 2015

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

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

Formula

E.g.f.: log(1-x)^2 + 2*x*log(1-x) + x^2.

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!.

a(n) = 2*(n-2)!*((n-1)*(Psi(n) + gamma) - n), n>2. - Vladeta Jovovic, Sep 21 2003