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

Author

Ran Pan, Feb 03 2016

Keywords

• nonn

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.

Crossrefs

Cf. A000108, A001628, A033184.

Programs

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 *)

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 */

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