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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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10
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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(1/k,k=1..n). - 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
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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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Vincenzo Librandi, Table of n, a(n) for n = 0..200
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 83
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FORMULA
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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( 2*k*|S1(n-1,k)|, k=1..n-1) = 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
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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);
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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 *)
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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))))};
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CROSSREFS
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Equals 2 * A000254(n+1), n>0.
Equals, for n=>2, the second right hand column of A028421.
Sequence in context: A009468 A088819 A177478 * A245119 A012270 A009585
Adjacent sequences: A052514 A052515 A052516 * A052518 A052519 A052520
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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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Typos in Maple program fixed by Johannes W. Meijer, Oct 16 2009
Name improved by Geoffrey Critzer, Jun 13 2013
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STATUS
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approved
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