OFFSET
0,3
COMMENTS
See Dziemianczuk (2014) for precise definition.
LINKS
Lars Blomberg, Table of n, a(n) for n = 0..100
M. Dziemianczuk, On Directed Lattice Paths With Additional Vertical Steps, arXiv preprint arXiv:1410.5747 [math.CO], 2014.
M. Dziemianczuk, On Directed Lattice Paths With Additional Vertical Steps, Discrete Mathematics, Volume 339, Issue 3, 6 March 2016, Pages 1116-1139.
FORMULA
G.f.: P2(x) = (1-x*P1(x))/(1-x-x*P1(x)), where P1(x) = 2*(1-x)/(3*x) - (2*sqrt(1-5*x-2*x^2)/(3*x))*sin(Pi/6 + arccos((20*x^3-6*x^2+15*x-2)/(2*(1-5*x-2*x^2)^(3/2)))/3). - See Dziemianczuk (2014), Proposition 11.
a(n) = A260771(n-1), n > 0 [see Proof of Proposition 11]. - R. J. Mathar, Aug 02 2015
a(n) = (1/n)*Sum_{j=0..floor((n-1)/4)} (-1)^j*C(n,j)*C(3*n-4*j-2,n-4*j-1), a(0)=1. - Vladimir Kruchinin, Apr 04 2019
PROG
(Maxima)
a(n):=if n=0 then 1 else sum((-1)^j*binomial(n, j)*binomial(3*n-4*j-2, n-4*j-1), j, 0, floor((n-1)/4))/n; /* Vladimir Kruchinin, Apr 04 2019 */
CROSSREFS
KEYWORD
nonn
AUTHOR
N. J. A. Sloane, Jul 30 2015
EXTENSIONS
More terms from Lars Blomberg, Aug 01 2015
STATUS
approved