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

0,5

Links

G. C. Greubel, Table of n, a(n) for n = 0..445

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 84

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

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 *)

Prog

(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
(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
(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

Crossrefs

Cf. A000254, A000276.

Sequence in context: A026077 A348601 A065113 * A135032 A122074 A289208

Adjacent sequences: A052515 A052516 A052517 * A052519 A052520 A052521

Keyword

easy,nonn

Author

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

Status

approved