A268447 Number of North-East lattice paths from (0,0) to (n,n) that cross the diagonal y = x horizontally exactly four times.

1, 18, 189, 1518, 10350, 63180, 356265, 1893294, 9612108, 47071640, 223926516, 1040310648, 4739192952, 21238169904, 93865125915, 409972529754, 1772528290407, 7596549816030, 32308782859535, 136496564854650, 573285572389530, 2395339717603140, 9962435643667605

Offset

8,2

Comments

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

Links

Table of n, a(n) for n=8..30.

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

Formula

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

Conjecture: -(n+10)*(n-8)*a(n) +2*n*(2*n+1)*a(n-1)=0. - R. J. Mathar, Jun 07 2016

Mathematica

Rest[Rest[Rest[Rest[Rest[Rest[Rest[Rest[CoefficientList[Series[(2 (-1 + Sqrt[1 - 4 x] + 2 x)^4) / (1 + Sqrt[1 - 4 x] - 2 x)^5, {x, 0, 33}], x]]]]]]]]] (* Vincenzo Librandi, Feb 06 2016 *)

Prog

(PARI) x='x+O('x^100); Vec((2*(-1 + (1 - 4*x)^(1/2) + 2*x)^4)/(1 + (1 - 4*x)^(1/2) - 2*x)^5) \\ Altug Alkan, Feb 04 2016

Crossrefs

Cf. A268446.

Sequence in context: A073385 A036219 A022646 * A259163 A004314 A338099

Adjacent sequences: A268444 A268445 A268446 * A268448 A268449 A268450

Keyword

nonn

Author

Ran Pan, Feb 04 2016

Status

approved