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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. 2
0, 4, 12, 82, 454, 2912, 18652, 124299, 841400, 5800725, 40506816, 286137616, 2040430976, 14670243774, 106225269954, 773958961125, 5670067999156, 41742291894425, 308645064367896, 2291123920091484, 17067970534656790 (list; graph; refs; listen; history; text; internal format)
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-Fran├žois 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: A010370 A197852 A305334 * A263866 A208802 A226960

Adjacent sequences:  A205334 A205335 A205336 * A205338 A205339 A205340

KEYWORD

nonn

AUTHOR

R. H. Hardin, Jan 26 2012

STATUS

approved

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Last modified February 27 13:18 EST 2020. Contains 332306 sequences. (Running on oeis4.)