A174347 Expansion of (1 - 2*x - sqrt(1 - 8*x + 8*x^2))/(2*x*(1-x)).

1, 3, 11, 47, 223, 1135, 6063, 33535, 190399, 1103231, 6497407, 38779647, 234043647, 1425869567, 8757326591, 54163521535, 337060285439, 2108928587775, 13258969458687, 83720567447551, 530692157964287, 3375836610256895

Offset

0,2

Comments

Binomial transform of large Schroeder numbers A006318.

Hankel transform is 2^binomial(n+1,2).

Series reversion of (-1)^(n+1)*A001333(n). - Vladimir Reshetnikov, Nov 08 2015

Series reversion of x + 3*x^2 + 11*x^3 + ... is x - 3*x^2 + 7*x^3 - ... - Michael Somos, Nov 09 2015

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Paul Barry, Laurent Biorthogonal Polynomials and Riordan Arrays, arXiv preprint arXiv:1311.2292 [math.CA], 2013.

Aoife Hennessy, A Study of Riordan Arrays with Applications to Continued Fractions, Orthogonal Polynomials and Lattice Paths, Ph. D. Thesis, Waterford Institute of Technology, Oct. 2011.

Formula

G.f.: 1/(1-x-2x/(1-x/(1-x-2x/(1-x/(1-x-2x/(1-x/(1-... (continued fraction);

a(n) = Sum_{k=0..n} binomial(n,k)*A006318(k).

D-finite with recurrence: (n+1)*a(n) + 3*(1-3n)*a(n-1) + 4*(4n-5)*a(n-2) + 8(2-n)*a(n-3) = 0. - R. J. Mathar, Dec 08 2011

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

0 = a(n)*(+64*a(n+1) - 224*a(n+2) + 192*a(n+3) - 32*a(n+4)) + a(n+1)*(-32*a(n+1) + 208*a(n+2) - 260*a(n+3) + 52*a(n+4)) + a(n+2)*(-12*a(n+2) + 61*a(n+3) - 21*a(n+4)) + a(n+3)*(+3*a(n+3) + a(n+4)) for all n>=0. - Michael Somos, Nov 09 2015

Example

G.f. = 1 + 3*x + 11*x^2 + 47*x^3 + 223*x^4 + 1135*x^5 + 6063*x^6 + 33535*x^7 + ...

Mathematica

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

Prog

(PARI) x='x+O('x^35); Vec((1-2*x-sqrt(1-8*x+8*x^2))/(2*x*(1-x))) \\ Altug Alkan, Nov 08 2015
(Magma) m:=35; R<x>:=PowerSeriesRing(Rationals(), m); Coefficients(R!((1-2*x-Sqrt(1-8*x+8*x^2))/(2*x*(1-x)))); // G. C. Greubel, Sep 22 2018

Crossrefs

Cf. A006318, A001333.

Sequence in context: A301409 A295539 A359120 * A062146 A216947 A090365

Adjacent sequences: A174344 A174345 A174346 * A174348 A174349 A174350

Keyword

easy,nonn

Author

Paul Barry, Mar 16 2010

Status

approved