OFFSET
0,2
COMMENTS
Equivalent to nonnegative walks from (0,0) to (7*n,0) with step set [1,2], [1,-5].
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^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)
The sequence defined by b(n) := [x^n] A(x)^n begins [1, 3, 161, 9804, 630401, 41789278, 2824792568, 193553976353, ...] and conjecturally satisfies the congruence b(p) == b(1) (mod p^3) for prime p >= 11 (checked up to p = 101). - Peter Bala, Sep 14 2021
a(n) ~ c * 7^(7*n) / (n^(3/2) * 2^(2*n) * 5^(5*n)), where c = 0.0538519123304380623474844037127876191519207214308040151922885271364215631... = s*sqrt((3 - 2*s + 2*s^2 - s^3 + s^4 + 6*r*s^7 - 2*r*s^8 + 4*r*s^9 + 3*r^2*s^14) / (63 - 56*s + 72*s^2 - 45*s^3 + 55*s^4 + 273*r*s^7 - 105*r*s^8 + 240*r*s^9 + 210*r^2*s^14)) / (2*sqrt(Pi)), where r = 12500/823543 and s = 1.129379978325... is the root of the equation -16807 + 24010*s - 13720*s^2 + 7350*s^3 - 3500*s^4 + 1250*s^5 = 0. - Vaclav Kotesovec, Sep 16 2021
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-François Alcover, Dec 04 2018 *)
nmax = 20; CoefficientList[Series[Exp[Sum[Binomial[7*k, 2*k]*x^k/(7*k), {k, 1, nmax}]], {x, 0, nmax}], x] (* Vaclav Kotesovec, Sep 16 2021 *)
CROSSREFS
KEYWORD
nonn,walk
AUTHOR
Bryan T. Ek, Mar 04 2018
STATUS
approved