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A233481 Number of singletons (strong fixed points) in pair-partitions. 3
0, 1, 4, 21, 144, 1245, 13140, 164745, 2399040, 39834585, 742940100, 15374360925, 349484058000, 8654336615925, 231842662751700, 6679510641428625, 205916703920928000, 6762863294018456625, 235719416966063530500, 8689887736412502745125 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

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).

LINKS

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]

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

MAPLE

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

MATHEMATICA

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

PROG

(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(); [a.next() for i in range(17)]  # Peter Luschny, Dec 14 2013

CROSSREFS

Cf. A001147, A034430, A082590, A076729.

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

KEYWORD

nonn

AUTHOR

Wojciech Bozejko, Dec 11 2013

STATUS

approved

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Last modified December 14 22:42 EST 2019. Contains 329987 sequences. (Running on oeis4.)