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%I #33 Oct 06 2021 13:24:11
%S 1,4,178,11654,900239,76266406,6853777795,641688752961,61916364799849,
%T 6113859987916630,614832988424140624,62752222758863566993,
%U 6483650829899569496380,676834416167597357806799,71278487569046416052210050,7563527671079260544924794587,807900192360879042402313084390
%N 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.
%C Equivalent to nonnegative walks from (0,0) to (9*n,0) with step set [1,2], [1,-7].
%H M. T. L. Bizley, <a href="/A060941/a060941.pdf">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</a>, Journal of the Institute of Actuaries, Vol. 80, No. 1 (1954): 55-62. [Cached copy]
%H Bryan Ek, <a href="https://arxiv.org/abs/1803.10920">Lattice Walk Enumeration</a>, arXiv:1803.10920 [math.CO], 2018.
%H Bryan Ek, <a href="https://arxiv.org/abs/1804.05933">Unimodal Polynomials and Lattice Walk Enumeration with Experimental Mathematics</a>, arXiv:1804.05933 [math.CO], 2018.
%F 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.
%F From _Peter Bala_, Jan 03 2019: (Start)
%F O.g.f.: A(x) = exp( Sum_{n >= 1} (1/9)*binomial(9*n, 2*n)*x^n/n ) - Bizley. Cf. A274244.
%F 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)
%F The sequence defined by b(n) := [x^n] A(x)^n begins [1, 4, 372, 39298, 4384884, 504464254, 59183637186, 7038517648243, ...] 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
%F a(n) ~ c * 3^(18*n) / (n^(3/2) * 2^(2*n) * 7^(7*n)), where c = 0.0389180896257538883301359279112039841187646397413254619045749515282872957... - _Vaclav Kotesovec_, Sep 16 2021
%e For n=1, the possible walks are EEEEEEENN, EEEEEENEN, EEEEENEEN, EEEENEEEN.
%t terms = 17; f[_] = 0;
%t 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}];
%t CoefficientList[f[t], t] (* _Jean-François Alcover_, Dec 04 2018 *)
%t nmax = 20; CoefficientList[Series[Exp[Sum[Binomial[9*k, 2*k]*x^k/(9*k), {k, 1, nmax}]], {x, 0, nmax}], x] (* _Vaclav Kotesovec_, Sep 16 2021 *)
%Y Cf. A001764, A060941, A300386, A300388, A300389, A274244.
%K nonn,walk
%O 0,2
%A _Bryan T. Ek_, Mar 04 2018
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