A260771 Certain directed lattice paths.

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

0,2

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.

Formula

G.f.: P0(x) = 1/(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) = Sum_{m=0..n+1} ((-1)^(n-m+1)*binomial(n+1,m) * Sum_{i=m..n+m} (binomial(i-1,i-m) * Sum_{j=0..n-m+1} (binomial(j,n+m-j-i)*2^(-n+m+2*j+i)*3^(n-m-j+1)*binomial(n-m+1,j))))/(n+1). - Vladimir Kruchinin, Feb 28 2016

a(n) ~ c * (22 + 10*sqrt(5))^(n/2) / n^(3/2), where c = 1/sqrt((5/2 - sqrt(5) + sqrt(85*sqrt(5)-190))*Pi) = 0.7820861193303307654051... . - Vaclav Kotesovec, Feb 28 2016

Mathematica

Table[Sum[(-1)^(n-l+1)*Binomial[n+1, l] * Sum[Binomial[i-1, i-l] * Sum[Binomial[j, n+l-j-i] * 2^(-n+l+2*j+i) * 3^(n-l-j+1) * Binomial[n-l+1, j], {j, 0, n-l+1}], {i, l, n+l}], {l, 0, n+1}]/(n+1), {n, 0, 20}] (* Vaclav Kotesovec, Feb 28 2016, after Vladimir Kruchinin *)

Prog

(Maxima) a(n):=sum((-1)^(n-l+1)*binomial(n+1, l)*sum(binomial(i-1, i-l)*sum(binomial(j, n+l-j-i)*2^(-n+l+2*j+i)*3^(n-l-j+1)*binomial(n-l+1, j), j, 0, n-l+1), i, l, n+l), l, 0, n+1)/(n+1); /* Vladimir Kruchinin, Feb 28 2016 */

Crossrefs

Sequence in context: A371432 A366089 A368936 * A260773 A368937 A174796

Adjacent sequences: A260768 A260769 A260770 * A260772 A260773 A260774

Keyword

nonn

Author

N. J. A. Sloane, Jul 30 2015

Extensions

More terms from Lars Blomberg, Aug 01 2015

Status

approved