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

Author

N. J. A. Sloane, Jul 30 2015

Keywords

• nonn

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.

Programs

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 */

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

Extensions

More terms from Lars Blomberg, Aug 01 2015