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A268447
Number of North-East lattice paths from (0,0) to (n,n) that cross the diagonal y = x horizontally exactly four times.
0
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
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
KEYWORD
nonn
AUTHOR
Ran Pan, Feb 04 2016
STATUS
approved