A268446 Number of North-East lattice paths from (0,0) to (n,n) that cross the diagonal y = x horizontally exactly three times.

1, 14, 119, 798, 4655, 24794, 123970, 592020, 2731365, 12271350, 53993940, 233646504, 997490844, 4211628008, 17620076360, 73153696336, 301758997386, 1237956266316, 5054988087457, 20558563992050, 83322650532485, 336691526641470, 1356968880100470, 5456577564869340, 21898107332699325

Offset

6,2

Comments

It is related to paired pattern P_3 in Section 3.3 in Pan and Remmel's link.

Links

G. C. Greubel, Table of n, a(n) for n = 6..1000

Daniel Birmajer, Juan B. Gil, Michael D. Weiner, Bounce statistics for rational lattice paths, arXiv:1707.09918 [math.CO], 2017, p. 9.

Ran Pan, Jeffrey B. Remmel, Paired patterns in lattice paths, arXiv:1601.07988 [math.CO], 2016.

Formula

G.f.: -((2*(2*x + f(x) - 1)^3)/(-2*x + f(x) +1)^4), where f(x) = sqrt(1 - 4*x).

From Karol A. Penson, Nov 19 2016: (Start)

a(n) = 14*binomial(2*n+1, n-6)/(n+8).

G.f.: 4^7*z^6/(1+sqrt(1-4*z))^14. - shifted by Georg Fischer, Feb 13 2020

E.g.f.(in Maple notation): hypergeom([7,15/2],[1,15],4*z).

Recurrence: 2*(n+1)*(2*n+3)*a(n)-(n-5)*(n+9)*a(n+1)=0. - Georg Fischer, Feb 13 2020

Asymptotics: (114688*n-6838272)*4^n*sqrt(1/n)/(sqrt(Pi)*n^2). (End)

Maple

a:= n-> 14*binomial(2*n+1, n-6)/(n+8): seq(a(n), n=6..23);

Mathematica

Rest[Rest[Rest[Rest[Rest[Rest[CoefficientList[Series[-((2 (2 x + Sqrt[1 - 4 x] - 1)^3) / (-2 x + Sqrt[1 - 4 x] + 1)^4), {x, 0, 33}], x]]]]]]] (* Vincenzo Librandi, Feb 06 2016 *) (* or *)
RecurrenceTable[{2*(n+1)*(2*n+3)*a[n]-(n-5)*(n+9)*a[n+1]==0, a[6]==1}, a, {n, 6, 25}] (* Georg Fischer, Feb 13 2020 *)

Prog

(PARI) for(n=6, 25, print1(14*binomial(2*n+1, n-6)/(n+8), ", ")) \\ G. C. Greubel, Apr 09 2017, shifted by Georg Fischer, Feb 13 2020

Crossrefs

Sequence in context: A023012 A073383 A022642 * A221230 A240051 A202072

Adjacent sequences: A268443 A268444 A268445 * A268447 A268448 A268449

Keyword

nonn

Author

Ran Pan, Feb 04 2016

Status

approved