OFFSET
0,3
LINKS
Alois P. Heinz, Rows n = 0..60, flattened
Wikipedia, Counting lattice paths
EXAMPLE
/\
T(3,-1) = 1: / \/\
.
/\
/ \ /\/\
T(3,0) = 3: / \ / \ /\/\/\
.
/\
T(3,1) = 1: /\/ \
.
Triangle T(n,k) begins:
: 1 ;
: 1 ;
: 2 ;
: 1, 3, 1 ;
: 1, 3, 6, 3, 1 ;
: 2, 5, 8, 12, 8, 5, 2 ;
: 1, 4, 9, 16, 22, 28, 22, 16, 9, 4, 1 ;
: 1, 4, 11, 21, 34, 49, 60, 69, 60, 49, 34, 21, 11, 4, 1 ;
MAPLE
b:= proc(x, y, v) option remember; expand(
`if`(min(y, v, x-max(y, v))<0, 0, `if`(x=0, 1, (l-> add(add(
b(x-1, y+i, v+j)*z^((y-v)/2+(i-j)/4), i=l), j=l))([-1, 1]))))
end:
T:= n-> (p-> seq(coeff(p, z, i), i=ldegree(p)..degree(p)))(
add(b(n, (n-2*j)$2), j=0..n/2)):
seq(T(n), n=0..12);
MATHEMATICA
b[x_, y_, v_] := b[x, y, v] = Expand[If[Min[y, v, x - Max[y, v]] < 0, 0, If[x == 0, 1, Function[l, Sum[Sum[b[x - 1, y + i, v + j] z^((y - v)/2 + (i - j)/4), {i, l}], {j, l}]][{-1, 1}]]]];
T[n_] := Function[p, Table[Coefficient[p, z, i], {i, Range[Exponent[p, z, Reverse @@ # &], Exponent[p, z]]}]][Sum[b[n, n-2j, n-2j], {j, 0, n/2}]];
Table[T[n], {n, 0, 12}] // Flatten (* Jean-François Alcover, May 31 2018, from Maple *)
CROSSREFS
KEYWORD
nonn,tabf
AUTHOR
Alois P. Heinz, Mar 02 2018
STATUS
approved