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

%I

%S 1,5,25,155,1025,7167,51945,387000,2944860,22791189,178840639,

%T 1419569398,11377983292,91957314063,748575327757,6132254500856,

%U 50514620902564,418174191239443,3477075679541185,29026557341147912,243184916545458556

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

%C Column 4 of A204213.

%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,-1,0,1,2,3,4}. - _David Nguyen_, Dec 16 2016

%H R. H. Hardin, <a href="/A204209/b204209.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_{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

%e Some solutions for n=5

%e ..0....0....0....0....0....0....0....0....0....0....0....0....0....0....0....0

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

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

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

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

%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, j] Binomial[-9j + 5i - 1, 4i - 9j] (-1)^j, {j, 0, (4i)/9}]) a[n - i], {i, 1, n}]/n];

%t a /@ Range[1, 21] (* _Jean-Fran├žois Alcover_, Sep 24 2019, after _Vladimir Kruchinin_ *)

%o (Maxima)

%o 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 */

%K nonn

%O 1,2

%A _R. H. Hardin_, Jan 12 2012

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Last modified June 6 14:04 EDT 2020. Contains 334827 sequences. (Running on oeis4.)