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A052517 Number of ordered pairs of cycles over all n-permutations having two cycles. 20
0, 0, 2, 6, 22, 100, 548, 3528, 26136, 219168, 2053152, 21257280, 241087680, 2972885760, 39605518080, 566931294720, 8678326003200, 141468564787200, 2446811181158400, 44753976117043200, 863130293635276800 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

a(n) is a function of the harmonic numbers. If we discard the first term and set a(0)=0, a(1)=2..then a(n) = 2n!*h(n) where h(n) = Sum_{k=1..n} 1/k. - Gary Detlefs, Aug 04 2010

a(n+1) is twice the sum over all permutations of the number of its cycles (fixed points included). - Olivier Gérard, Oct 23 2012

In a game where n differently numbered cards are drawn in a random sequence, and a point is earned every time the newest card is either the highest or the lowest yet drawn (the first card gets two points as it is both the highest and the lowest), the expected number of points earned is a(n+1)/n!, for instance if n=3, there are two ways of getting 3 points and four ways of getting 4 points, giving an average of 22/6 = 3 2/3. - Elliott Line, Mar 19 2020

REFERENCES

G. Boole, A Treatise On The Calculus of Finite Differences, Dover, 1960, p. 30.

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..200

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 83

FORMULA

E.g.f.: (log(1 - x))^2. - Michael Somos, Feb 05 2004

a(n) ~ 2*(n-1)!*log(n)*(1+gamma/log(n)), where gamma is Euler-Mascheroni constant (A001620). - Vaclav Kotesovec, Oct 08 2012

a(n) = Sum_{k=1..n-1} 2*k*|S1(n-1,k)| = 2*|S1(n,2)|. - Olivier Gérard, Oct 23 2012

0 = a(n) * n^2 - a(n+1) * (2*n+1) + a(n+2) for all n in Z. - Michael Somos, Apr 23 2014

0 = a(n)*(a(n+1) - 7*a(n+2) + 6*a(n+3) - a(n+4)) + a(n+1)*(a(n+1) - 6*a(n+2) + 4*a(n+3)) + a(n+2)*(-3*a(n+2)) if n>0. - Michael Somos, Apr 23 2014

For n>=2, a(n) = (n-2)! * Sum_{i=1..n-1} Sum_{j=1..n-1} (i+j)/(i*j). - Pedro Caceres, Feb 14 2021

EXAMPLE

a(3)=6 because we have the ordered pairs of cycles: ((1)(23)), ((23)(1)), ((2)(13)), ((13)(2)), ((3)(12)), ((12)(3)). - Geoffrey Critzer, Jun 13 2013

G.f. = 2*x^2 + 6*x^3 + 22*x^4 + 100*x^5 + 548*x^6 + 3528*x^7 + ...

MAPLE

pairsspec := [S, {S=Prod(B, B), B=Cycle(Z)}, labeled]: seq(combstruct[count](pairsspec, size=n), n=0..20); # Typos fixed by Johannes W. Meijer, Oct 16 2009

MATHEMATICA

Flatten[{0, Table[(n+1)!*Sum[1/(k*(n+1-k)), {k, 1, n}], {n, 0, 20}]}] (* Vaclav Kotesovec, Oct 08 2012 *)

With[{m = 25}, CoefficientList[Series[Log[1-x]^2, {x, 0, m}], x]*Range[0, m]!] (* G. C. Greubel, May 13 2019 *)

PROG

(PARI) {a(n) = if( n<0, 0, n! * sum(k=1, n-1, 1 / (k * (n - k))))};

(MAGMA) m:=25; R<x>:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!( Log(1-x)^2 )); [0, 0] cat [Factorial(n+1)*b[n]: n in [1..m-2]]; // G. C. Greubel, May 13 2019

(Sage) m = 25; T = taylor(log(1-x)^2, x, 0, m); [factorial(n)*T.coefficient(x, n) for n in (0..m)] # G. C. Greubel, May 13 2019

CROSSREFS

Equals 2 * A000254(n+1), n>0.

Equals, for n=>2, the second right hand column of A028421.

Cf. A302827, A304589.

Sequence in context: A009468 A088819 A177478 * A245119 A012270 A009585

Adjacent sequences:  A052514 A052515 A052516 * A052518 A052519 A052520

KEYWORD

easy,nonn

AUTHOR

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

EXTENSIONS

Name improved by Geoffrey Critzer, Jun 13 2013

STATUS

approved

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Last modified April 20 06:59 EDT 2021. Contains 343125 sequences. (Running on oeis4.)