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 A300388 The number of paths of length 11*n from the origin to the line y = 2*x/9 with unit East and North steps that stay below the line or touch it. 5
 1, 5, 345, 35246, 4255288, 563796161, 79264265868, 11612106079203, 1753402118587333, 270965910076404428, 42648418241303137766, 6813002989827352100145, 1101807202785456951146158, 180034116076502209139781574, 29677341363243548521326632028, 4929368173228370040701922315332 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS Equivalent to nonnegative walks from (0,0) to (11*n,0) with step set  [1,2], [1,-9]. LINKS M. T. L. Bizley, Derivation of a new formula for the number of minimal lattice paths from (0, 0) to (km, kn) having just t contacts with the line my = nx and having no points above this line; and a proof of Grossman's formula for the number of paths which may touch but do not rise above this line, Journal of the Institute of Actuaries, Vol. 80, No. 1 (1954): 55-62. [Cached copy] Bryan Ek, Lattice Walk Enumeration, arXiv:1803.10920 [math.CO], 2018. Bryan Ek, Unimodal Polynomials and Lattice Walk Enumeration with Experimental Mathematics, arXiv:1804.05933 [math.CO], 2018. FORMULA G.f. satisfies: f = f^55*t^5 + 4*f^46*t^4 - f^45*t^4 + 5*f^44*t^4 + 6*f^37*t^3 - 3*f^36*t^3 + 12*f^35*t^3 - 4*f^34*t^3 + 10*f^33*t^3 + 4*f^28*t^2 - 3*f^27*t^2 + 9*f^26*t^2 - 6*f^25*t^2 + 12*f^24*t^2 - 6*f^23*t^2 + 10*f^22*t^2 + f^19*t - f^18*t + 2*f^17*t - 2*f^16*t + 3*f^15*t - 3*f^14*t + 4*f^13*t - 4*f^12*t + 5*f^11*t + 1. From Peter Bala, Jan 03 2019: (Start) O.g.f.: A(x) = exp( Sum_{n >= 1} (1/11)*binomial(11*n, 2*n)*x^n/n ) - Bizley. Cf. A274256. Recurrence: a(0) = 1 and a(n) = (1/n) * Sum_{k = 0..n-1} (1/11)*binomial(11*n-11*k, 2*n-2*k)*a(k) for n >= 1. (End) EXAMPLE For n=1, the walks are EEEEEEEEENN, EEEEEEEENEN, EEEEEEENEEN, EEEEEENEEEN, EEEEENEEEEN. MATHEMATICA terms = 16; f[_] = 0; Do[f[t_] = f[t]^55 t^5 + 4 f[t]^46 t^4 - f[t]^45 t^4 + 5 f[t]^44 t^4 + 6 f[t]^37 t^3 - 3 f[t]^36 t^3 + 12 f[t]^35 t^3 - 4 f[t]^34 t^3 + 10 f[t]^33 t^3 + 4 f[t]^28 t^2 - 3 f[t]^27 t^2 + 9 f[t]^26 t^2 - 6 f[t]^25 t^2 + 12 f[t]^24 t^2 - 6 f[t]^23 t^2 + 10 f[t]^22 t^2 + f[t]^19 t - f[t]^18 t + 2 f[t]^17 t - 2 f[t]^16 t + 3 f[t]^15 t - 3 f[t]^14 t + 4 f[t]^13 t - 4 f[t]^12 t + 5 f[t]^11 t + 1 + O[t]^terms, {terms}]; CoefficientList[f[t], t] (* Jean-François Alcover, Dec 04 2018 *) CROSSREFS Cf. A001764, A060941, A300386, A300387, A300389, A274256. Sequence in context: A203684 A124477 A059839 * A269556 A227448 A210820 Adjacent sequences:  A300385 A300386 A300387 * A300389 A300390 A300391 KEYWORD nonn,walk AUTHOR Bryan T. Ek, Mar 04 2018 STATUS approved

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Last modified August 12 03:34 EDT 2020. Contains 336436 sequences. (Running on oeis4.)