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A233481
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Number of singletons (strong fixed points) in pair-partitions.
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4
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0, 1, 4, 21, 144, 1245, 13140, 164745, 2399040, 39834585, 742940100, 15374360925, 349484058000, 8654336615925, 231842662751700, 6679510641428625, 205916703920928000, 6762863294018456625, 235719416966063530500, 8689887736412502745125
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OFFSET
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0,3
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COMMENTS
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For h(V) = number of singletons (non-crossing chords) in the pair-partition of 2n-elementary set P_2(2n), let T(2n) = sum_{V in P_2(2n)} h(V).
Elements of the sequence a(n) = T(2n).
a(n) is the number of linear chord diagrams on 2n vertices with one marked chord such that none of the remaining n-1 chords cross the marked chord, see [Young]. - Donovan Young, Aug 11 2020
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LINKS
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G. C. Greubel, Table of n, a(n) for n = 0..400
Marek Bozejko, Wojciech Bozejko, Generalized Gaussian processes and relations with random matrices and positive definite functions on permutation groups, (2013), arXiv:1301.2502 [math.PR]
Donovan Young, A critical quartet for queuing couples, arXiv:2007.13868 [math.CO], 2020.
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FORMULA
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a(n) = T_{2n} = n*sum_{k=0..(n-1)} (2k-1)!!*(2n-2k-1)!!, where (2n-1)!! = 1*3*5*...*(2n-1).
From Peter Luschny, Dec 16 2013: (Start)
E.g.f.: x/((1-x)*sqrt(1-2*x)).
a(n) = 2*n*Gamma(1/2+n)*2_F_1([1/2,-n+1],[3/2],-1)/sqrt(Pi), where 2_F_1 is the hypergeometric function.
a(n) = n*((3*n-4)*a(n-1)/(n-1)-(2*n-3)*a(n-2)) for n>1.
a(n) = n*A034430(n-1) for n>=1.
a(n+1)/(n+1)! = JacobiP(n, 1/2, -n-1, 3).
2^n*a(n+1)/(n+1)! = A082590(n).
2^n*a(n+1)/(n+1) = A076729(n). (End)
a(n) ~ 2^(n+1/2) * n^n / exp(n). - Vaclav Kotesovec, Dec 20 2013
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MAPLE
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a := n -> 2*n*GAMMA(1/2+n)*hypergeom([1/2, -n+1], [3/2], -1)/sqrt(Pi); seq(round(evalf(a(n), 32)), n=0..17); # Peter Luschny, Dec 16 2013
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MATHEMATICA
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Table[Sum[(2 k - 1)!! (2 n - 2 k - 1)!!, {k, 0, n - 1}], {n, 0, 30}] (* T. D. Noe, Dec 13 2013 *)
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PROG
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(Sage)
def A233481():
a, b, n = 0, 1, 1
while True:
yield a
n += 1
a, b = b, n*((3*n-4)*b/(n-1)-(2*n-3)*a)
a = A233481(); [next(a) for i in range(17)] # Peter Luschny, Dec 14 2013
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CROSSREFS
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Cf. A001147, A034430, A082590, A076729, A336598.
A081054 counts pair-partitions of a fixed size without singletons, i.e., linear chord diagrams with 2n nodes and n arcs in which each arc crosses another arc.
Sequence in context: A228063 A228111 A305986 * A308337 A307525 A327872
Adjacent sequences: A233478 A233479 A233480 * A233482 A233483 A233484
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KEYWORD
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nonn
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AUTHOR
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Wojciech Bozejko, Dec 11 2013
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STATUS
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approved
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