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

1,2

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

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.

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

Example

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

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 *)

Prog

(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 */

Crossrefs

Cf. A205341.

Sequence in context: A197852 A362384 A305334 * A359047 A263866 A208802

Adjacent sequences: A205334 A205335 A205336 * A205338 A205339 A205340

Keyword

nonn

Author

R. H. Hardin, Jan 26 2012

Status

approved