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

0, 1, 4, 21, 144, 1245, 13140, 164745, 2399040, 39834585, 742940100, 15374360925, 349484058000, 8654336615925, 231842662751700, 6679510641428625, 205916703920928000, 6762863294018456625, 235719416966063530500, 8689887736412502745125

Offset: 0

Author

Wojciech Bozejko, Dec 11 2013

Keywords

• nonn

Comments

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

Links

• G. C. Greubel, Table of n, a(n) for n = 0..400
• Marek Bozejko and Wojciech Bozejko, Generalized Gaussian processes and relations with random matrices and positive definite functions on permutation groups, arXiv:1301.2502 [math.PR], 2013.
• Donovan Young, A critical quartet for queuing couples, arXiv:2007.13868 [math.CO], 2020.

Crossrefs

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.

Programs

Maple

a := n -> 2*n*GAMMA(1/2+n)*hypergeom([1/2,-n+1],[3/2],-1)/sqrt(Pi);
seq(simplify(a(n)), n = 0..19); # Peter Luschny, Dec 16 2013
# Alternative:
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

Mathematica

Table[Sum[(2 k - 1)!! (2 n - 2 k - 1)!!, {k, 0, n - 1}], {n,0,30}] (* T. D. Noe, Dec 13 2013 *)

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

Formula

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

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