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A268446 Number of North-East lattice paths from (0,0) to (n,n) that cross the diagonal y = x horizontally exactly three times. 2
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 (list; graph; refs; listen; history; text; internal format)
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

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Last modified January 28 05:27 EST 2022. Contains 350654 sequences. (Running on oeis4.)