A268400 Number of North-East lattice paths from (0,0) to (n,n) that bounce off the diagonal y = x to the right exactly twice.

1, 5, 23, 99, 413, 1691, 6842, 27464, 109631, 435887, 1728018, 6835668, 26996393, 106486529, 419639903, 1652533719, 6504159137, 25589302163, 100646529977, 395775842389, 1556107102849, 6117771240251, 24050813530815, 94550689834203, 371715533473021, 1461430355605367, 5746128800657639, 22594839306797223

Offset

3,2

Comments

This sequence is related to paired pattern P_2 in Pan and Remmel's link.

Links

G. C. Greubel, Table of n, a(n) for n = 3..1000

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

Formula

G.f.: -((-1 + f(x))^3*x*(-1 + f(x) + 2*x))/(2*(1 - f(x) + (-5 + f(x))*x)^3), where f(x) = sqrt(1 - 4*x).

a(n) = ((Sum_{m=1..n-3}((m+2)*(Sum_{k=0..m/2}((m^2+(3-2*k)*m+k^2-3*k+2)*binomial(m-k,k)))*binomial(2*n-m-5,n-m-3)))+2*binomial(2*n-4,n-2))/(2*n-2), n>2. - Vladimir Kruchinin, Feb 28 2016

a(n) ~ 7*2^(2*n+2)/(sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Feb 28 2016

D-finite with recurrence (n-1)*(9*n-56)*a(n) +(-31*n^2+203*n-16)*a(n-1) +(-193*n^2+1371*n-2138)*a(n-2) +3*(217*n^2-1497*n+2274)*a(n-3) +2*(41*n-94)*(2*n-5)*a(n-4)=0. - R. J. Mathar, Jul 24 2022

Mathematica

Rest[Rest[Rest[CoefficientList[Series[-((-1 + Sqrt[1 - 4 x])^3 x (-1 + Sqrt[1-4 x] + 2 x))/(2 (1 - Sqrt[1 - 4 x] + (-5 + Sqrt[1 - 4 x]) x)^3), {x, 0, 40}], x]]]] (* Vincenzo Librandi, Feb 28 2016 *)

Prog

(Maxima)
a(n):=((sum((m+2)*(sum((m^2+(3-2*k)*m+k^2-3*k+2)*binomial(m-k, k), k, 0, m/2)) *binomial(2*n-m-5, n-m-3), m, 1, n-3))+2*binomial(2*n-4, n-2))/(2*n-2); /* Vladimir Kruchinin, Feb 28 2016 */

Crossrefs

Cf. A000108, A001628, A033184.

Sequence in context: A215038 A084615 A181331 * A364754 A339232 A196489

Adjacent sequences: A268397 A268398 A268399 * A268401 A268402 A268403

Keyword

nonn

Author

Ran Pan, Feb 03 2016

Status

approved