A105696 Expansion of (1-x)/sqrt((1-3*x)/(1+x)).

1, 1, 2, 6, 16, 44, 122, 342, 966, 2746, 7846, 22514, 64836, 187288, 542438, 1574666, 4580400, 13347324, 38956182, 113861922, 333226560, 976353876, 2863756158, 8407877394, 24707200854, 72663608178, 213864889770, 629893319902

Offset: 0

Author

Paul Barry, Apr 17 2005

Keywords

• easy
• nonn

Comments

Apply the Riordan array (1-x,x/(1+x)) to C(2n,n).

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000

Crossrefs

Cf. A002426.

Programs

Mathematica

CoefficientList[Series[(1-x)/Sqrt[(1-3x)/(1+x)],{x,0,30}],x] (* Harvey P. Dale, Feb 12 2016 *)

PARI

x='x+O('x^50); Vec((1-x)/sqrt((1-3*x)/(1+x))) \\ G. C. Greubel, Mar 02 2017

Formula

a(n) = A002426(n) - A002426(n-2).

a(n) = Sum_{k=0..floor(n/2)} ( 2*C(n-2, k)*C(n-k, k) ) - C(1, n).

Conjecture D-finite with recurrence: n*a(n) +(-3*n+2)*a(n-1) +(-n+2)*a(n-2) +3*(n-4)*a(n-3)=0. - R. J. Mathar, Jan 22 2020

a(n) ~ 4 * 3^(n - 3/2) / sqrt(Pi*n). - Vaclav Kotesovec, Nov 19 2021