A024483 a(n) = binomial(2*n, n) mod binomial(2*n-2, n-1).

0, 2, 10, 42, 168, 660, 2574, 10010, 38896, 151164, 587860, 2288132, 8914800, 34767720, 135727830, 530365050, 2074316640, 8119857900, 31810737420, 124718287980, 489325340400, 1921133836440, 7547311500300, 29667795388452, 116686713634848

Offset: 2

Author

Clark Kimberling

Keywords

• nonn
• easy

Comments

Apart from its root term -1: central terms of the triangle in A051631: a(n) = A051631(2*(n-1), n-1). - Reinhard Zumkeller, Nov 13 2011

Define an array m(i,j) by m(1,j)=m(j,1)=j*(j+1)/2 for j=0,1,2,3,... and m(i,j) = m(i,j-1) + m(i-1,j+1); the diagonal m(k,k) for k=1,2,3... gives the numbers in this sequence. - J. M. Bergot, May 02 2012

The central terms of triangle A051631 (including the root term -1) are given by (n-1)*(n+1)*Gamma(2*n+1)/Gamma(n+2)^2 with n >= 0. - Peter Luschny, Nov 24 2013

Index the sequence from n=0 so that a(0)=1, a(1)=0, a(2)=2, a(3)=10, ... a(n) is the number of walks using steps U=(1,1) and D=(1,-1) from the origin to (2n,0) that rise above and dip below the x axis. a(2) = 2 because we have: DUUD and UDDU. - Geoffrey Critzer, Jan 11 2014

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 2..1000

Crossrefs

Cf. A000108, A000984, A001622, A051631.

Programs

Haskell

a024483 n = a051631 (2*(n-1)) (n-1) -- Reinhard Zumkeller, Nov 13 2011

Maple

seq((n-1)*binomial(2*n, n)/(n+1), n=1..25); # Zerinvary Lajos, Feb 28 2007

Mathematica

nn=20; d=(1-(1-4x)^(1/2))/(2x); Drop[CoefficientList[Series[1/(1-2x d)-2(d-1), {x,0,nn}],x],1] (* Geoffrey Critzer, Jan 11 2014 *)
Table[Mod[Binomial[2 n, n], Binomial[2 n - 2, n - 1]], {n, 2, 26}] (* Michael De Vlieger, Sep 13 2016 *)

Sage

def a(n): return n*(n-2)*factorial(2*(n-1))/factorial(n)^2
[a(n) for n in (2..26)]  # Peter Luschny, Nov 24 2013

Formula

a(n) = ((n-2)/n)*binomial(2*n-2, n-1) = (n-2)*A000108(n-1). - Vladeta Jovovic, Aug 03 2002

a(n) = 2*binomial(2n-3, n-3) = 2*A002054(n-2). - Ralf Stephan, Jan 15 2004

a(n) = A000984(n-1) - 2*A000108(n-1). - Geoffrey Critzer, Jan 11 2014

a(n) ~ 4^(n-1)/sqrt(Pi*n). - Ilya Gutkovskiy, Sep 13 2016

D-finite with recurrence n*a(n) +(-7*n+8)*a(n-1) +6*(2*n-5)*a(n-2)=0. - R. J. Mathar, Apr 27 2020

From Amiram Eldar, Mar 24 2022: (Start)

Sum_{n>=3} 1/a(n) = 5/6 - Pi/(9*sqrt(3)).

Sum_{n>=3} (-1)^(n+1)/a(n) = 26*sqrt(5)*log(phi)/25 - 7/10, where phi is the golden ratio (A001622). (End)

Extensions

More terms from Zerinvary Lajos, Oct 02 2007