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A300387 The number of paths of length 9*n from the origin to the line y = 2*x/7 with unit East and North steps that stay below the line or touch it. 5
1, 4, 178, 11654, 900239, 76266406, 6853777795, 641688752961, 61916364799849, 6113859987916630, 614832988424140624, 62752222758863566993, 6483650829899569496380, 676834416167597357806799, 71278487569046416052210050, 7563527671079260544924794587, 807900192360879042402313084390 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

Equivalent to nonnegative walks from (0,0) to (9*n,0) with step set [1,2], [1,-7].

LINKS

Table of n, a(n) for n=0..16.

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^36*t^4+3*f^29*t^3-f^28*t^3+4*f^27*t^3+3*f^22*t^2-2*f^21*t^2+6*f^20*t^2-3*f^19*t^2+6*f^18*t^2+f^15*t-f^14*t+2*f^13*t-2*f^12*t+3*f^11*t-3*f^10*t+4*f^9*t+1.

From Peter Bala, Jan 03 2019: (Start)

O.g.f.: A(x) = exp( Sum_{n >= 1} (1/9)*binomial(9*n, 2*n)*x^n/n ) - Bizley. Cf. A274244.

Recurrence: a(0) = 1 and a(n) = (1/n) * Sum_{k = 0..n-1} (1/9)*binomial(9*n-9*k, 2*n-2*k)*a(k) for n >= 1. (End)

EXAMPLE

For n=1, the possible walks are EEEEEEENN, EEEEEENEN, EEEEENEEN, EEEENEEEN.

MATHEMATICA

terms = 17; f[_] = 0;

Do[f[t_] = f[t]^36 t^4 + 3 f[t]^29 t^3 - f[t]^28 t^3 + 4 f[t]^27 t^3 + 3 f[t]^22 t^2 - 2 f[t]^21 t^2 + 6 f[t]^20 t^2 - 3 f[t]^19 t^2 + 6 f[t]^18 t^2 + f[t]^15 t - f[t]^14 t + 2 f[t]^13 t - 2 f[t]^12 t + 3 f[t]^11 t - 3 f[t]^10 t + 4 f[t]^9 t + 1 + O[t]^terms, {terms}];

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

CROSSREFS

Cf. A001764, A060941, A300386, A300388, A300389, A274244.

Sequence in context: A210780 A127606 A041945 * A082393 A176351 A330771

Adjacent sequences:  A300384 A300385 A300386 * A300388 A300389 A300390

KEYWORD

nonn,walk

AUTHOR

Bryan T. Ek, Mar 04 2018

STATUS

approved

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Last modified September 30 21:27 EDT 2020. Contains 337440 sequences. (Running on oeis4.)