A186338 Expansion of 1/(1-2x/(1-2x/(1-x/(1-2x/(1-2x/(1-x/(1-2x/(1-... (continued fraction).

1, 2, 8, 36, 172, 860, 4460, 23820, 130268, 726236, 4112972, 23599724, 136906748, 801671996, 4732110828, 28128179276, 168222049052, 1011509012636, 6111445499532, 37084090264364, 225899543897916, 1380918157453052, 8468524718133804, 52085281291575052

Offset: 0

Author

Paul Barry, Feb 18 2011

Keywords

• nonn
• easy

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..200
• Xiaomei Chen, Yuan Xiang, Counting generalized Schröder paths, arXiv:2009.04900 [math.CO], 2020.

Crossrefs

Hankel transform is A186339.

Programs

Mathematica

CoefficientList[Series[(Sqrt[1-10*x+25*x^2-16*x^3]+3*x-1)/(2*x*(2*x-1)), {x, 0, 20}], x] (* Vaclav Kotesovec, Oct 24 2012 *)

Maxima

a(n):=sum(sum(binomial(j+1, i)*binomial(2*j-i, j-i)*binomial(n-j+i-1, n-j), i, 0, j)/(j+1)*2^(n-j), j, 0, n); /* Vladimir Kruchinin, Jan 25 2020 */

Formula

G.f.: (sqrt(1-10x+25x^2-16x^3)+3x-1)/(2x(2x-1)).

Conjecture: (n+1)*a(n) +3*(1-4n)*a(n-1) +15*(3n-4)*a(n-2) +6*(26-11n)*a(n-3) +16*(2n-7)*a(n-4)=0. - R. J. Mathar, Nov 17 2011

a(n) = Sum_{k, 0<=k<=n} A091866(n,k)*2^k. - Philippe Deléham, Nov 27 2011

a(n) ~ sqrt(7*sqrt(17)-17)*((9+sqrt(17))/2)^n/(2*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Oct 24 2012

From Vladimir Kruchinin, Jan 25 2020: (Start)

a(n) = Sum_{j=0..n} Sum_{i=0..j} C(j+1, i)*C(2*j-i, j-i)*C(n-j+i-1,n-j) /(j+1)*2^(n-j).

a(n) = Sum_{i=0..n-1} a(i)*(2^(n-i-1)+a(n-i-1)). (End)