%I #21 Sep 24 2019 12:37:16

%S 0,4,12,82,454,2912,18652,124299,841400,5800725,40506816,286137616,

%T 2040430976,14670243774,106225269954,773958961125,5670067999156,

%U 41742291894425,308645064367896,2291123920091484,17067970534656790

%N Number of length n+1 nonnegative integer arrays starting and ending with 0 with adjacent elements unequal but differing by no more than 4.

%C Column 4 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 {-4,-3,-2,-1,1,2,3,4}. - _David Nguyen_, Dec 20 2016

%H R. H. Hardin, <a href="/A205337/b205337.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..(4*(i-l))/9}((-1)^j*binomial(i-l,j)*binomial(-l+4*(-l-2*j+i)-j+i-1,4*(-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 ..2....3....2....4....4....4....1....2....4....3....3....1....2....3....2....4

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

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

%e ..2....2....1....1....3....4....1....4....4....2....4....2....4....3....4....2

%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 + 4(-l - 2j + i) - j + i - 1, 4(-l - 2j + i) - j], {j, 0, (4(i - l))/9}] (-1)^l, {l, 0, i}] a[n - i], {i, 1, n}]/n];

%t a /@ Range[1, 21] (* _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+4*(-l-2*j+i)-j+i-1,4*(-l-2*j+i)-j),j,0,(4*(i-l))/9)*(-1)^l,l,0,i)*a(n-i),i,1,n)/n; /* _Vladimir Kruchinin_, Apr 07 2017 */

%Y Cf. A205341.

%K nonn

%O 1,2

%A _R. H. Hardin_, Jan 26 2012