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

0,2

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.

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)

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 *)

Prog

(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 */

Crossrefs

Hankel transform is A186339.

Sequence in context: A330793 A352862 A109980 * A190862 A110837 A372088

Adjacent sequences: A186335 A186336 A186337 * A186339 A186340 A186341

Keyword

nonn,easy

Author

Paul Barry, Feb 18 2011

Status

approved