A205336 - OEIS

%I #19 Sep 24 2019 14:31:54

%S 0,3,6,35,138,689,3272,16522,83792,434749,2278888,12093271,64741330,

%T 349470487,1899418046,10387322922,57111322368,315523027610,

%U 1750681516380,9751416039535,54507046599094,305650440453943,1718956630038438

%N Number of length n+1 nonnegative integer arrays starting and ending with 0 with adjacent elements unequal but differing by no more than 3.

%C Column 3 of A205341.

%C Number of excursions (walks starting at the origin, ending on the x-axis, and never go below the x-axis in between) with n steps from {-3,-2,-1,1,2,3}. - _David Nguyen_, Dec 20 2016

%H R. H. Hardin, <a href="/A205336/b205336.txt">Table of n, a(n) for n = 1..210</a>

%H C. Banderier, C. Krattenthaler, A. Krinik, D. Kruchinin, V. Kruchinin, D. Nguyen, and M. Wallner, <a href="https://arxiv.org/abs/1609.06473">Explicit formulas for enumeration of lattice paths: basketball and the kernel method</a>, arXiv preprint arXiv:1609.06473 [math.CO], 2016.

%F a(n) = Sum_{i=1..n}((Sum_{l=0..i}(binomial(i,l)*(Sum_{j=0=(3*(i-l))/7}((-1)^j*binomial(i-l,j)*binomial(-l+3*(-l-2*j+i)-j+i-1,3*(-l-2*j+i)-j)))*(-1)^l))*a(n-i))/n, a(0)=1. - _Vladimir Kruchinin_, Apr 07 2017

%e Some solutions for n=5

%e ..0....0....0....0....0....0....0....0....0....0....0....0....0....0....0....0

%e ..3....3....3....1....3....1....1....3....3....2....1....3....1....3....3....3

%e ..4....6....2....0....2....3....3....2....5....4....4....1....3....2....2....0

%e ..2....5....5....3....4....4....2....3....4....1....2....2....0....4....0....2

%e ..3....2....2....2....2....1....3....2....2....3....1....1....2....3....2....1

%e ..0....0....0....0....0....0....0....0....0....0....0....0....0....0....0....0

%t a[n_] := a[n] = If[n == 0, 1, Sum[(Sum[Binomial[i, l] (Sum[(-1)^j Binomial[i - l, j] Binomial[-l + 3(-l - 2j + i) - j + i - 1, 3(-l - 2j + i) - j], {j, 0, (3(i - l))/7}]) (-1)^l, {l, 0, i}]) a[n - i], {i, 1, n}]/n];

%t a /@ Range[1, 23] (* _Jean-François Alcover_, Sep 24 2019, after _Vladimir Kruchinin_ *)

%o (Maxima)

%o a(n):=if n=0 then 1 else sum((sum(binomial(i,l)*(sum((-1)^j*binomial(i-l,j)*binomial(-l+3*(-l-2*j+i)-j+i-1,3*(-l-2*j+i)-j),j,0,(3*(i-l))/7))*(-1)^l,l,0,i))*a(n-i),i,1,n)/n; /* _Vladimir Kruchinin_, Apr 07 2017 */

%K nonn

%O 1,2

%A _R. H. Hardin_, Jan 26 2012