A166229 Expansion of (1-2x-sqrt(1-8x+8x^2))/(2x).

1, 2, 8, 36, 176, 912, 4928, 27472, 156864, 912832, 5394176, 32282240, 195264000, 1191825920, 7331457024, 45406194944, 282896763904, 1771868302336, 11150040870912, 70461597988864, 446971590516736, 2845144452292608

Offset

0,2

Comments

Binomial transform of A166228. Hankel transform is A166231.

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..300

Formula

a(n) = 0^n + Sum_{k = 0..n} C(n-1,k-1)*A006318(k). - Paul Barry, Nov 04 2009

G.f.: 1/(1-2x/(1-x-x/(1-2x/(1-x-x/(1-2x/(1-x-x/(1-... (continued fraction). - Paul Barry, Dec 10 2009

Recurrence: (n+1)*a(n) = 4*(2*n-1)*a(n-1) - 8*(n-2)*a(n-2). - Vaclav Kotesovec, Oct 20 2012

a(n) ~ sqrt(1+sqrt(2))*(4+2*sqrt(2))^n/(sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Oct 20 2012

From Peter Bala, May 01 2024: (Start)

O.g.f.: A(x) = x*S(x/(1 - x)), where S(x) = (1 - x - sqrt(1 - 6*x + x^2))/(2*x) is the o.g.f. for the large Schröder numbers A006318.

a(n) = A174347(n+1) - A174347(n).

The g.f. satisfies x^2*A(x)^2 - x*(1 - 2*x)*A(x) + x*(1 - x) = 0.

A(x) = (1 - x)/(1 - 2*x - x*(1 - x)/(1 - 2*x - x*(1 - x)/(1 - 2*x - ...))). (End)

Mathematica

CoefficientList[Series[(1-2*x-Sqrt[1-8*x+8*x^2])/(2*x), {x, 0, 20}], x] (* Vaclav Kotesovec, Oct 20 2012 *)

Crossrefs

Cf. A006318, A174347.

Sequence in context: A190862 A110837 A372088 * A109318 A113327 A227791

Adjacent sequences: A166226 A166227 A166228 * A166230 A166231 A166232

Keyword

easy,nonn

Author

Paul Barry, Oct 09 2009

Status

approved