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A026569 a(n)=T(n,n), T given by A026568. Also a(n) = number of integer strings s(0),...,s(n) counted by T, such that s(n)=0. 8
1, 1, 3, 5, 13, 27, 67, 153, 375, 893, 2189, 5319, 13089, 32155, 79479, 196573, 487833, 1212135, 3018355, 7525585, 18792303, 46980373, 117589689, 294613155, 738844719, 1854484305, 4658460165, 11710592711, 29458662005, 74151824271 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

Number of grand Motzkin n-paths avoiding UF. - David Scambler, Jun 20 2013

LINKS

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

J. W. Layman, The Hankel Transform and Some of its Properties, J. Integer Sequences, 4 (2001), #01.1.5.

FORMULA

a(n) = sum(k=0..floor(n/2), binomial(2*k, k)*binomial(n-k, k) ) - Paul Barry, Sep 09 2004

G.f.: sqrt(1/((1-x)*(1-x-4*x^2))). - Ralf Stephan, Jan 08 2004

a(n) = 1/n*((2*n-1)*a(n-1)+(3*n-3)*a(n-2)-(4*n-6)*a(n-3)). - Vladeta Jovovic, Mar 12 2005

a(n) = sum(k=0..n, C(k, n-k)*C(2*(n-k), n-k)). - Paul Barry, Jul 30 2005

G.f.: 1/(1-x-2x^2/(1-0x-x^2/(1-x-x^2/(1-0x-2x^2/(1-x-x^2/.... (continued fraction). Paul Barry, Dec 07 2008

Conjecture: n*a(n) +(1-2n)*a(n-1) +3*(1-n)*a(n-2) +2*(2n-3)*a(n-3) = 0. - R. J. Mathar, Nov 16 2011

a(n) ~ sqrt((5+13/sqrt(17))/8) * ((1+sqrt(17))/2)^n/sqrt(Pi*n). - Vaclav Kotesovec, Aug 10 2013

EXAMPLE

For a(3) = 5 the five grand Motzkin paths are FDU, DFU, FUD, UDF and FFF. The paths containing UF, namely UFD and DUF, are avoided. - David Scambler, Jun 20 2013

MATHEMATICA

CoefficientList[Series[Sqrt[1/((1-x)(1-x-4x^2))], {x, 0, 30}], x] (* Harvey P. Dale, Oct 06 2011 *)

CROSSREFS

Sequence in context: A223645 A190570 A000631 * A035082 A005198 A160823

Adjacent sequences:  A026566 A026567 A026568 * A026570 A026571 A026572

KEYWORD

nonn

AUTHOR

Clark Kimberling

STATUS

approved

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Last modified June 25 20:26 EDT 2017. Contains 288730 sequences.