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Number of singletons (strong fixed points) in pair-partitions.
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%I #65 Mar 14 2023 18:30:11

%S 0,1,4,21,144,1245,13140,164745,2399040,39834585,742940100,

%T 15374360925,349484058000,8654336615925,231842662751700,

%U 6679510641428625,205916703920928000,6762863294018456625,235719416966063530500,8689887736412502745125

%N Number of singletons (strong fixed points) in pair-partitions.

%C 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).

%C Elements of the sequence a(n) = T(2n).

%C 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

%H G. C. Greubel, <a href="/A233481/b233481.txt">Table of n, a(n) for n = 0..400</a>

%H Marek Bozejko and Wojciech Bozejko, <a href="http://arxiv.org/abs/1301.2502">Generalized Gaussian processes and relations with random matrices and positive definite functions on permutation groups</a>, arXiv:1301.2502 [math.PR], 2013.

%H Donovan Young, <a href="https://arxiv.org/abs/2007.13868">A critical quartet for queuing couples</a>, arXiv:2007.13868 [math.CO], 2020.

%F a(n) = T_{2n} = n*sum_{k=0..(n-1)} (2k-1)!!*(2n-2k-1)!!, where (2n-1)!! = 1*3*5*...*(2n-1).

%F From _Peter Luschny_, Dec 16 2013: (Start)

%F E.g.f.: x/((1-x)*sqrt(1-2*x)).

%F 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.

%F a(n) = n*((3*n-4)*a(n-1)/(n-1)-(2*n-3)*a(n-2)) for n>1.

%F a(n) = n*A034430(n-1) for n>=1.

%F a(n+1)/(n+1)! = JacobiP(n, 1/2, -n-1, 3).

%F 2^n*a(n+1)/(n+1)! = A082590(n).

%F 2^n*a(n+1)/(n+1) = A076729(n). (End)

%F a(n) ~ 2^(n+1/2) * n^n / exp(n). - _Vaclav Kotesovec_, Dec 20 2013

%F a(n) = (2*n)! * [z^(2*n)] 2*u*exp(u)*hypergeom([1/2], [3/2], u), where u = (z/2)^2. - _Peter Luschny_, Mar 14 2023

%p a := n -> 2*n*GAMMA(1/2+n)*hypergeom([1/2,-n+1],[3/2],-1)/sqrt(Pi);

%p seq(simplify(a(n)), n = 0..19); # _Peter Luschny_, Dec 16 2013

%p # Alternative:

%p u := (z/2)^2: egf := 2*u*exp(u)*hypergeom([1/2], [3/2], u): ser := series(egf, z, 40): seq((2*n)!*coeff(ser, z, 2*n), n = 0..19); # _Peter Luschny_, Mar 14 2023

%t Table[Sum[(2 k - 1)!! (2 n - 2 k - 1)!!, {k, 0, n - 1}], {n,0,30}] (* _T. D. Noe_, Dec 13 2013 *)

%o (Sage)

%o def A233481():

%o a, b, n = 0, 1, 1

%o while True:

%o yield a

%o n += 1

%o a, b = b, n*((3*n-4)*b/(n-1)-(2*n-3)*a)

%o a = A233481(); [next(a) for i in range(17)] # _Peter Luschny_, Dec 14 2013

%Y Cf. A001147, A034430, A082590, A076729, A336598.

%Y 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.

%K nonn

%O 0,3

%A _Wojciech Bozejko_, Dec 11 2013