OFFSET
1,2
COMMENTS
Column 3 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 {-3,-2,-1,1,2,3}. - 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=(3*(i-l))/7}((-1)^j*binomial(i-l,j)*binomial(-l+3*(-l-2*j+i)-j+i-1,3*(-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
..3....3....3....1....3....1....1....3....3....2....1....3....1....3....3....3
..4....6....2....0....2....3....3....2....5....4....4....1....3....2....2....0
..2....5....5....3....4....4....2....3....4....1....2....2....0....4....0....2
..3....2....2....2....2....1....3....2....2....3....1....1....2....3....2....1
..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 + 3(-l - 2j + i) - j + i - 1, 3(-l - 2j + i) - j], {j, 0, (3(i - l))/7}]) (-1)^l, {l, 0, i}]) a[n - i], {i, 1, n}]/n];
a /@ Range[1, 23] (* 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+3*(-l-2*j+i)-j+i-1, 3*(-l-2*j+i)-j), j, 0, (3*(i-l))/7))*(-1)^l, l, 0, i))*a(n-i), i, 1, n)/n; /* Vladimir Kruchinin, Apr 07 2017 */
CROSSREFS
KEYWORD
nonn
AUTHOR
R. H. Hardin, Jan 26 2012
STATUS
approved