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A233481 Number of singletons (strong fixed points) in pair-partitions. 3

%I

%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).

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

%H Marek Bozejko, 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>, (2013), arXiv:1301.2502 [math.PR]

%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

%p 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

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

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

%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

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Last modified January 18 05:09 EST 2020. Contains 330995 sequences. (Running on oeis4.)