A125558 Central column of triangle A090181.

1, 1, 6, 50, 490, 5292, 60984, 736164, 9202050, 118195220, 1551580888, 20734762776, 281248448936, 3863302870000, 53644719852000, 751920156592200, 10626401036545650, 151269944167296900, 2167317913508055000

Offset: 0

Author

Philippe Deléham, Jan 01 2007, Oct 11 2007

Keywords

• easy
• nonn

Comments

[1,6,50,490,5292,...] is a column in triangle of Narayana numbers A001263.

Number of Dyck 2n-paths with exactly n peaks. - Peter Luschny, May 10 2014

For n > 0, number of pairs of non-intersecting lattice paths with steps (1,0), (0,1), where one path goes from (0,0) to (n,n) and the other from (1,0) to (n+1,n). The proof is by switching intersecting path pairs after their first intersection, giving a(n) = binomial(2*n,n)^2 - binomial(2*n+1,n) * binomial(2*n-1,n). - Jeremy Tan, Apr 12 2021

Links

• Harvey P. Dale, Table of n, a(n) for n = 0..800

Crossrefs

Equals A000888(n)/2 for n>0.

Cf. A090181.

Programs

Maple

seq(ceil(1/2*(n+1)*((binomial(2*n,n)/(1+n))^2)), n=0..18); # Zerinvary Lajos, Jun 18 2007

Mathematica

CoefficientList[
Series[1 + (HypergeometricPFQ[{1/2, 1/2}, {2}, 16 x] - 1)/(2), {x, 0,
    20}], x]
Join[{1},Table[CatalanNumber[n]^2 (n+1)/2,{n,20}]] (* Harvey P. Dale, Oct 19 2011 *)

Formula

a(0)=1, a(n) = Catalan(n)^2*(n+1)/2 = A000108(n)^2*(n+1)/2 for n>0.

a(n) = A090181(2*n, n).

G.f.: 1 + x*3F2( 1, 3/2, 3/2; 2, 3;16 x) = 1 + ( 2F1( 1/2, 1/2; 2;16*x) - 1)/2. - Olivier Gérard, Feb 16 2011

D-finite with recurrence n*(n+1)*a(n) -4*(2*n-1)^2*a(n-1)=0. - R. J. Mathar, Feb 08 2021

a(n) = binomial(2*n,n)^2 - binomial(2*n+1,n) * binomial(2*n-1,n). - Jeremy Tan, Apr 12 2021