A260773 Certain directed lattice paths.

1, 1, 2, 7, 30, 142, 716, 3771, 20502, 114194, 648276, 3737270, 21819980, 128757020, 766680856, 4600866643, 27797553638, 168949310378, 1032267189636, 6336728149794, 39062959379620, 241720286906116, 1500910751651752, 9348824475860702, 58398701313158780

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

Sequence in context: A366089 A368936 A260771 * A368937 A174796 A046648

Adjacent sequences: A260770 A260771 A260772 * A260774 A260775 A260776

Keyword

nonn

Author

N. J. A. Sloane, Jul 30 2015

Extensions

More terms from Lars Blomberg, Aug 01 2015

Status

approved