A204208 Number of length n+1 nonnegative integer arrays starting and ending with 0 with adjacent elements differing by no more than 3.

1, 4, 16, 78, 404, 2208, 12492, 72589, 430569, 2596471, 15870357, 98102191, 612222083, 3852015239, 24408653703, 155629858911, 997744376239, 6427757480074, 41590254520410, 270163621543421, 1761179219680657

Offset: 1

Author

R. H. Hardin, Jan 12 2012

Keywords

• nonn

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

Examples

			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
		

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.

Programs

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 *)

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 */

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

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