A102881 Expansion of (1+x)/sqrt(1-4x^2-8x^3-4x^4).

1, 1, 2, 6, 12, 32, 80, 200, 520, 1336, 3472, 9072, 23744, 62432, 164544, 434688, 1150944, 3052768, 8110784, 21581120, 57498496, 153378048, 409583616, 1094848768, 2929288960, 7843943680, 21020501504, 56371941888, 151276652544

Offset: 0

Author

Paul Barry, Jan 15 2005

Keywords

• easy
• nonn

Comments

Binomial transform is A101500. Binomial transform of A102882.

Apparently the number of grand Motzkin paths of length n that avoid FF (double flat steps). - David Scambler, Jul 04 2013

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Programs

Mathematica

CoefficientList[Series[(1+x)/Sqrt[1-4*x^2-8*x^3-4*x^4], {x, 0, 20}], x] (* Vaclav Kotesovec, Feb 08 2014 *)

Formula

G.f.: (1+x)/sqrt((1-2x-2x^2)(1+2x+2x^2)).

D-finite with recurrence: n*a(n) +(n-2)*a(n-1) +4*(-n+1)*a(n-2) +12*(-n+2)*a(n-3) +12*(-n+3)*a(n-4) +4*(-n+4)*a(n-5)=0. - R. J. Mathar, Nov 16 2012

D-finite with recurrence (of order 4): (n-1)*n*a(n) = 4*(n-1)^2*a(n-2) + 4*(n-2)*(2*n-1)*a(n-3) + 4*(n-3)*n*a(n-4). - Vaclav Kotesovec, Feb 08 2014

a(n) ~ sqrt(54+30*sqrt(3)) * (1+sqrt(3))^n / (12 * sqrt(Pi*n)). - Vaclav Kotesovec, Feb 08 2014