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 A268462 Expansion of (2 x^4*(5 - 12*x + 8*x^2))/(1 - 2*x)^4. 6
 0, 0, 0, 0, 10, 56, 224, 768, 2400, 7040, 19712, 53248, 139776, 358400, 901120, 2228224, 5431296, 13074432, 31129600, 73400320, 171573248, 397934592, 916455424, 2097152000, 4771020800, 10796138496, 24310185984, 54492397568, 121634816000, 270448721920, 599147937792 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,5 COMMENTS a(n) is the number of North-East lattice paths from (0,0) to (n,n) in which total number of east steps below y = x-1 or above y = x+1 is exactly three. Details can be found in Section 4.1 in Pan and Remmel's link. LINKS Colin Barker, Table of n, a(n) for n = 0..1000 Ran Pan, Jeffrey B. Remmel, Paired patterns in lattice paths, arXiv:1601.07988 [math.CO], 2016. Index entries for linear recurrences with constant coefficients, signature (8,-24,32,-16). FORMULA G.f.: (2 x^4*(5 - 12*x + 8*x^2))/(1 - 2*x)^4. a(n) = 8*a(n-1)-24*a(n-2)+32*a(n-3)-16*a(n-4) for n>3. - Vincenzo Librandi, Feb 05 2016 a(n) = 2^(n-4)*(n-3)*(n+1)*(n+2)/3 for n>2. - Colin Barker, Feb 08 2016 MATHEMATICA CoefficientList[Series[(2 x^4 (5 - 12 x + 8 x^2)) / (1 - 2 x)^4, {x, 0, 33}], x] (* Vincenzo Librandi, Feb 05 2016 *) LinearRecurrence[{8, -24, 32, -16}, {0, 0, 0, 0, 10, 56, 224}, 40] (* Harvey P. Dale, Feb 10 2022 *) PROG (Magma) I:=[0, 0, 0, 0, 10, 56, 224]; [n le 7 select I[n] else 8*Self(n-1)-24*Self(n-2)+32*Self(n-3)-16*Self(n-4): n in [1..40]]; // Vincenzo Librandi, Feb 05 2016 (PARI) concat(vector(4), Vec(2*x^4*(5-12*x+8*x^2)/(1-2*x)^4 + O(x^100))) \\ Colin Barker, Feb 08 2016 CROSSREFS Sequence in context: A137931 A053493 A198833 * A296918 A001786 A258478 Adjacent sequences: A268459 A268460 A268461 * A268463 A268464 A268465 KEYWORD nonn,easy AUTHOR Ran Pan, Feb 04 2016 STATUS approved

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Last modified June 10 14:12 EDT 2023. Contains 363205 sequences. (Running on oeis4.)