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
Rui Duarte and António Guedes de Oliveira, Generating functions of lattice paths, Univ. do Porto (Portugal 2023).
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
D-finite with recurrence: 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-2*x^2/(1-0*x-x^2/(1-x-x^2/(1-0*x-2*x^2/(1-x-x^2/.... (continued fraction). Paul Barry, Dec 07 2008
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 *)
PROG
(PARI) my(x='x+O('x^30)); Vec( 1/sqrt((1-x)*(1-x-4*x^2)) ) \\ G. C. Greubel, Aug 03 2019
(Magma) R<x>:=PowerSeriesRing(Rationals(), 30); Coefficients(R!( 1/Sqrt((1-x)*(1-x-4*x^2)) )); // G. C. Greubel, Aug 03 2019
(Sage) (1/sqrt((1-x)*(1-x-4*x^2))).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, Aug 03 2019
(GAP) List([0..30], n-> Sum([0..Int(n/2)], k-> Binomial(2*k, k)*Binomial( n-k, k) )); # G. C. Greubel, Aug 03 2019
CROSSREFS
KEYWORD
nonn
AUTHOR
STATUS
approved