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

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.

Programs

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 *)

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 */

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

Extensions

More terms from Lars Blomberg, Aug 01 2015