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A300390 The number of paths of length 7*n from the origin to the line y = 3*x/4 with unit east and north steps that stay below the line or touch it. 1
1, 5, 227, 15090, 1182187, 101527596, 9247179818, 877362665128, 85783306955099, 8582893111512001, 874542924575207352, 90437361732467946334, 9467275300762187682554, 1001309098267187214993056, 106836493655355495755649544, 11485688815900189437990930096, 1242964338344397490958154292155 (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 [1,3], [1,-4].

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^35*t^5 - f^31*t^4 + f^30*t^4 - f^29*t^4 + 5*f^28*t^4 - f^25*t^3 + f^24*t^3 + 3*f^23*t^3 - 4*f^22*t^3 + 10*f^21*t^3 + f^19*t^2 - f^18*t^2 + 5*f^17*t^2 + 3*f^16*t^2 - 6*f^15*t^2 + 10*f^14*t^2 + f^13*t - f^12*t + 3*f^10*t + f^9*t - 4*f^8*t + 5*f^7*t + 1.

From Peter Bala, Jan 03 2019: (Start)

O.g.f.: A(x) = exp( Sum_{n >= 1} (1/7)*binomial(7*n, 3*n)*x^n/n ) - Bizley.

Recurrence: a(0) = 1 and a(n) = (1/n) * Sum_{k = 0..n-1} (1/7)*binomial(7*n-7*k, 3*n-3*k)*a(k) for n >= 1. (End)

EXAMPLE

For n=1, the possible walks are EEEENNN, EEENENN, EENEENN, EEENNEN, EENENEN.

MATHEMATICA

m = 17; f = 0; Do[f = f^35*t^5 - f^31*t^4 + f^30*t^4 - f^29*t^4 + 5*f^28*t^4 - f^25*t^3 + f^24*t^3 + 3*f^23*t^3 - 4*f^22*t^3 + 10*f^21*t^3 + f^19*t^2 - f^18*t^2 + 5*f^17*t^2 + 3*f^16*t^2 - 6*f^15*t^2 + 10*f^14*t^2 + f^13*t - f^12*t + 3*f^10*t + f^9*t - 4*f^8*t + 5*f^7*t + 1 + O[t]^m, {m}]; CoefficientList[f, t] (* Jean-Fran├žois Alcover, Feb 18 2019 *)

CROSSREFS

Cf. A001764, A060941, A300386, A300387, A300388, A300389, A300391.

Sequence in context: A200988 A201491 A181584 * A002142 A103732 A065757

Adjacent sequences:  A300387 A300388 A300389 * A300391 A300392 A300393

KEYWORD

nonn,walk,easy

AUTHOR

Bryan T. Ek, Mar 05 2018

STATUS

approved

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