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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. 3
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 (list; graph; refs; listen; history; text; internal format)
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

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 * A196489 A049674 A077277

Adjacent sequences:  A268397 A268398 A268399 * A268401 A268402 A268403

KEYWORD

nonn

AUTHOR

Ran Pan, Feb 03 2016

STATUS

approved

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Last modified September 26 06:54 EDT 2017. Contains 292502 sequences.