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A300386 The number of paths of length 7*n from the origin to the line y = 2*x/5 with unit East and North steps that stay below the line or touch it. 5
1, 3, 76, 2803, 121637, 5782513, 291437249, 15297882929, 827402061954, 45790180469312, 2580588279994441, 147592910517101281, 8544927937132306600, 499811636639428519226, 29491983283370728013309, 1753398440591481772556798, 104933899400256659634374549, 6316334518803437568442071134 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

Equivalent to nonnegative walks from (0,0) to (7*n,0) with step set  [1,2], [1,-5].

LINKS

Table of n, a(n) for n=0..17.

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^21*t^3+2*f^16*t^2-f^15*t^2+3*f^14*t^2+f^11*t-f^10*t+2*f^9*t-2*f^8*t+3*f^7*t+1.

From Peter Bala, Jan 02 2019: (Start)

O.g.f.: A(x) = exp( Sum_{n >= 1} (1/7)*binomial(7*n, 2*n)*x^n/n ) - Bizley. Cf. A274052.

Recurrence: a(0) = 1 and a(n) = (1/n) * Sum_{k = 0..n-1} (1/7)*binomial(7*n-7*k, 2*n-2*k)*a(k) for n >= 1. (End)

EXAMPLE

For n=1, the possible walks are EEEEENN, EEEENEN, EEENEEN.

MATHEMATICA

terms = 18; f[_] = 0;

Do[f[t_] = f[t]^21 t^3 + 2 f[t]^16 t^2 - f[t]^15 t^2 + 3 f[t]^14 t^2 + f[t]^11 t - f[t]^10 t + 2 f[t]^9 t - 2 f[t]^8 t + 3 f[t]^7 t + 1 + O[t]^terms, {terms}];

CoefficientList[f[t], t] (* Jean-Fran├žois Alcover, Dec 04 2018 *)

CROSSREFS

Cf. A001764, A060941, A300387, A300388, A300389, A274052.

Sequence in context: A042267 A201428 A141103 * A054747 A302375 A232030

Adjacent sequences:  A300383 A300384 A300385 * A300387 A300388 A300389

KEYWORD

nonn,walk

AUTHOR

Bryan T. Ek, Mar 04 2018

STATUS

approved

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Last modified September 18 02:19 EDT 2020. Contains 337164 sequences. (Running on oeis4.)