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A052517
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Number of ordered pairs of cycles over all n-permutations having two cycles.
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20
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0, 0, 2, 6, 22, 100, 548, 3528, 26136, 219168, 2053152, 21257280, 241087680, 2972885760, 39605518080, 566931294720, 8678326003200, 141468564787200, 2446811181158400, 44753976117043200, 863130293635276800
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OFFSET
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0,3
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COMMENTS
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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
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REFERENCES
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G. Boole, A Treatise On The Calculus of Finite Differences, Dover, 1960, p. 30.
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LINKS
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FORMULA
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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
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EXAMPLE
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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 + ...
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MAPLE
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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
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MATHEMATICA
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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 *)
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PROG
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(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
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CROSSREFS
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Equals, for n=>2, the second right hand column of A028421.
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KEYWORD
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easy,nonn
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AUTHOR
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encyclopedia(AT)pommard.inria.fr, Jan 25 2000
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EXTENSIONS
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STATUS
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approved
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