A204209 Number of length n+1 nonnegative integer arrays starting and ending with 0 with adjacent elements differing by no more than 4.

1, 5, 25, 155, 1025, 7167, 51945, 387000, 2944860, 22791189, 178840639, 1419569398, 11377983292, 91957314063, 748575327757, 6132254500856, 50514620902564, 418174191239443, 3477075679541185, 29026557341147912, 243184916545458556

Offset

1,2

Comments

Column 4 of A204213.

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,-1,0,1,2,3,4}. - David Nguyen, Dec 16 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_{j=0..(4*i)/9} (binomial(i,j)*binomial(-9*j+5*i-1,4*i-9*j)*(-1)^j))*a(n-i))/n. - Vladimir Kruchinin, Apr 06 2017

Example

Some solutions for n=5
..0....0....0....0....0....0....0....0....0....0....0....0....0....0....0....0
..3....3....3....2....1....2....4....4....3....3....2....2....0....1....4....4
..0....2....5....1....3....0....2....2....2....5....1....0....3....5....3....6
..0....1....6....2....4....3....1....3....3....2....2....1....3....3....1....3
..3....3....3....2....3....3....3....4....2....3....0....1....3....1....2....0
..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, j] Binomial[-9j + 5i - 1, 4i - 9j] (-1)^j, {j, 0, (4i)/9}]) 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, j)*binomial(-9*j+5*i-1, 4*i-9*j)*(-1)^j, j, 0, (4*i)/9))*a(n-i), i, 1, n)/n; /* Vladimir Kruchinin, Apr 06 2017 */

Crossrefs

Sequence in context: A216689 A297589 A092166 * A121112 A090014 A249475

Adjacent sequences: A204206 A204207 A204208 * A204210 A204211 A204212

Keyword

nonn

Author

R. H. Hardin, Jan 12 2012

Status

approved