A205337 Number of length n+1 nonnegative integer arrays starting and ending with 0 with adjacent elements unequal but differing by no more than 4.

0, 4, 12, 82, 454, 2912, 18652, 124299, 841400, 5800725, 40506816, 286137616, 2040430976, 14670243774, 106225269954, 773958961125, 5670067999156, 41742291894425, 308645064367896, 2291123920091484, 17067970534656790

Offset: 1

Author

R. H. Hardin, Jan 26 2012

Keywords

• nonn

Comments

Column 4 of A205341.

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

Examples

			Some solutions for n=5
..0....0....0....0....0....0....0....0....0....0....0....0....0....0....0....0
..2....3....2....4....4....4....1....2....4....3....3....1....2....3....2....4
..3....5....6....3....0....5....0....4....6....1....5....0....3....1....0....2
..6....1....2....2....1....3....3....6....3....4....3....1....6....2....1....5
..2....2....1....1....3....4....1....4....4....2....4....2....4....3....4....2
..0....0....0....0....0....0....0....0....0....0....0....0....0....0....0....0
		

Links

• R. H. Hardin, Table of n, a(n) for n = 1..210
• C. Banderier, C. Krattenthaler, A. Krinik, D. Kruchinin, V. Kruchinin, D. Nguyen, and M. Wallner, Explicit formulas for enumeration of lattice paths: basketball and the kernel method, arXiv preprint arXiv:1609.06473 [math.CO], 2016.

Crossrefs

Cf. A205341.

Programs

Mathematica

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];
a /@ Range[1, 21] (* Jean-François Alcover, Sep 24 2019, after Vladimir Kruchinin *)

Maxima

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 */

Formula

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