cp's OEIS Frontend

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

%I A233481 #65 Mar 14 2023 18:30:11
%S A233481 0,1,4,21,144,1245,13140,164745,2399040,39834585,742940100,
%T A233481 15374360925,349484058000,8654336615925,231842662751700,
%U A233481 6679510641428625,205916703920928000,6762863294018456625,235719416966063530500,8689887736412502745125
%N A233481 Number of singletons (strong fixed points) in pair-partitions.
%C A233481 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 A233481 Elements of the sequence a(n) = T(2n).
%C A233481 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 A233481 G. C. Greubel, <a href="/A233481/b233481.txt">Table of n, a(n) for n = 0..400</a>
%H A233481 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 A233481 Donovan Young, <a href="https://arxiv.org/abs/2007.13868">A critical quartet for queuing couples</a>, arXiv:2007.13868 [math.CO], 2020.
%F A233481 a(n) = T_{2n} = n*sum_{k=0..(n-1)} (2k-1)!!*(2n-2k-1)!!, where (2n-1)!! = 1*3*5*...*(2n-1).
%F A233481 From _Peter Luschny_, Dec 16 2013: (Start)
%F A233481 E.g.f.: x/((1-x)*sqrt(1-2*x)).
%F A233481 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 A233481 a(n) = n*((3*n-4)*a(n-1)/(n-1)-(2*n-3)*a(n-2)) for n>1.
%F A233481 a(n) = n*A034430(n-1) for n>=1.
%F A233481 a(n+1)/(n+1)! = JacobiP(n, 1/2, -n-1, 3).
%F A233481 2^n*a(n+1)/(n+1)! = A082590(n).
%F A233481 2^n*a(n+1)/(n+1) = A076729(n). (End)
%F A233481 a(n) ~ 2^(n+1/2) * n^n / exp(n). - _Vaclav Kotesovec_, Dec 20 2013
%F A233481 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 A233481 a := n -> 2*n*GAMMA(1/2+n)*hypergeom([1/2,-n+1],[3/2],-1)/sqrt(Pi);
%p A233481 seq(simplify(a(n)), n = 0..19); # _Peter Luschny_, Dec 16 2013
%p A233481 # Alternative:
%p A233481 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 A233481 Table[Sum[(2 k - 1)!! (2 n - 2 k - 1)!!, {k, 0, n - 1}], {n,0,30}] (* _T. D. Noe_, Dec 13 2013 *)
%o A233481 (Sage)
%o A233481 def A233481():
%o A233481     a, b, n = 0, 1, 1
%o A233481     while True:
%o A233481         yield a
%o A233481         n += 1
%o A233481         a, b = b, n*((3*n-4)*b/(n-1)-(2*n-3)*a)
%o A233481 a = A233481(); [next(a) for i in range(17)]  # _Peter Luschny_, Dec 14 2013
%Y A233481 Cf. A001147, A034430, A082590, A076729, A336598.
%Y A233481 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 A233481 nonn
%O A233481 0,3
%A A233481 _Wojciech Bozejko_, Dec 11 2013