OFFSET
1,2
COMMENTS
Column 3 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 {-3,-2,-1,0,1,2,3}. - 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
G.f.: exp( Sum_{n>=1} A025012(n)*x^n/n ) - 1, where A025012(n) = central coefficient of (1+x+x^2+x^3+x^4+x^5+x^6)^n. - Paul D. Hanna, Aug 01 2013
a(n) = Sum_{i=1..n}((Sum_{j=0..(3*i)/7}(binomial(i,j)*binomial(-7*j+4*i-1,3*i-7*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
..2....1....3....3....2....2....1....2....0....0....2....3....0....3....1....2
..5....3....2....2....2....3....1....5....3....0....2....4....3....2....0....3
..2....6....3....4....0....1....0....6....5....1....0....6....5....2....2....5
..2....3....3....3....2....3....3....3....2....1....0....3....3....0....3....3
..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[-7j + 4i - 1, 3i - 7j] (-1)^j, {j, 0, (3i)/7}]) a[n - i], {i, 1, n}]/n];
a /@ Range[1, 21] (* Jean-François Alcover, Sep 24 2019, after Vladimir Kruchinin *)
PROG
(PARI) {A025012(n)=polcoeff((1+x+x^2+x^3+x^4+x^5+x^6 +x*O(x^(3*n)))^n, 3*n)}
{a(n)=polcoeff(exp(sum(m=1, n, A025012(m)*x^m/m)+x*O(x^n)), n)}
for(n=0, 30, print1(a(n), ", ")) \\ Paul D. Hanna, Aug 01 2013
(Maxima)
a(n):=if n=0 then 1 else sum((sum(binomial(i, j)*binomial(-7*j+4*i-1, 3*i-7*j)*(-1)^j, j, 0, (3*i)/7))*a(n-i), i, 1, n)/n; /* Vladimir Kruchinin, Apr 06 2017 */
CROSSREFS
KEYWORD
nonn
AUTHOR
R. H. Hardin, Jan 12 2012
STATUS
approved