OFFSET
0,3
LINKS
Alois P. Heinz, Table of n, a(n) for n = 0..300
Wikipedia, Counting lattice paths
MAPLE
b:= proc(n, k, j) option remember; `if`(n=j, 1, add(
b(n-j, k, i)*(binomial(j-1, i-1)+binomial(i, k)
*binomial(j-1, i-1-k)), i=1..min(j+k, n-j)))
end:
a:= n-> 1 + add(b(n, j$2), j=1..n-1):
seq(a(n), n=0..33);
MATHEMATICA
b[n_, k_, j_] := b[n, k, j] = If[n==j, 1, Sum[b[n-j, k, i]*(Binomial[j-1, i - 1] + Binomial[i, k]*Binomial[j-1, i-1-k]), {i, 1, Min[j+k, n-j]}]];
a[n_] := 1 + Sum[b[n, j, j], {j, 1, n - 1}];
Table[a[n], {n, 0, 33}] (* Jean-François Alcover, May 31 2018, from Maple *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Alois P. Heinz, Jun 05 2017
STATUS
approved