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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
 1, 5, 25, 155, 1025, 7167, 51945, 387000, 2944860, 22791189, 178840639, 1419569398, 11377983292, 91957314063, 748575327757, 6132254500856, 50514620902564, 418174191239443, 3477075679541185, 29026557341147912, 243184916545458556 (list; graph; refs; listen; history; text; internal format)
 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

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Last modified March 28 07:59 EDT 2020. Contains 333079 sequences. (Running on oeis4.)