OFFSET
0,2
COMMENTS
Hankel transform is A165928. - Paul Barry, Sep 30 2009
Number of skew Dyck paths of semilength n with the down steps coming in two colors. - David Scambler, Jun 21 2013
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..200
C. Coker, Enumerating a class of lattice paths, Discrete Math., 271 (2003), 13-28.
J. Machacek, Lattice walks ending on a coordinate hyperlane avoiding backtracking and repeats, arXiv:2105.02417 [math.CO], 2021. See Thm. 4.4 G(x,E^1).
FORMULA
a(n) = 2*A059231(n), if n>0.
G.f.: (1 - x - sqrt((1 - x) * (1 - 9*x))) / (4*x) = 2 / (1 + sqrt((1 - 9*x) / (1 - x))) =: y satisfies 0 = (1 - x) * (1 - y) + 2*x*y^2. - Michael Somos, Mar 06 2004
Moment representation: a(n) = (1/(4*Pi))*Integral_{x=1..9} x^n*sqrt(-x^2+10x-9)/x+(1/2)*0^n. - Paul Barry, Sep 30 2009
D-finite with recurrence Recurrence: (n+1)*a(n) = 5*(2*n-1)*a(n-1) - 9*(n-2)*a(n-2). - Vaclav Kotesovec, Oct 14 2012
a(n) ~ 3^(2*n+1)/(2*sqrt(2*Pi)*n^(3/2)). - Vaclav Kotesovec, Oct 14 2012
G.f.: 1/(2*x) - 1/2 + G(0) where G(k) = 1 - 1/(x + x/(1 + 1/G(k+1) )) ; (continued fraction,3-step). - Sergei N. Gladkovskii, Nov 29 2012
EXAMPLE
G.f. = 1 + 2*x + 10*x^2 + 58*x^3 + 370*x^4 + 2514*x^5 + 17850*x^6 + 130890*x^7 + ...
MATHEMATICA
Table[SeriesCoefficient[2/(1+Sqrt[(1-9*x)/(1-x)]), {x, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Oct 14 2012 *)
PROG
(PARI) {a(n) = if( n<0, 0, polcoeff( 2 / (1 + sqrt((1 - 9*x) / (1 - x) + x * O(x^n))), n))}; /* Michael Somos, Mar 06 2004 */
(PARI) {a(n) = if( n<1, n==0, n++; 2 * polcoeff( serreverse( x * (1 - 4*x) / (1 - 3*x) + x * O(x^n)), n))}; /* Michael Somos, Mar 06 2004 */
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
N. J. A. Sloane, Sep 16 2003
EXTENSIONS
More terms from Ray Chandler, Sep 17 2003
STATUS
approved