A300322 Number T(n,k) of Dyck paths of semilength n such that 2*k is the difference between the area under the right half of the path and the area under the left half of the path; triangle T(n,k), n>=0, -floor(n*(n-1)/6) <= k <= floor(n*(n-1)/6), read by rows.
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, 2, 7, 15, 31, 53, 82, 114, 147, 171, 186, 171, 147, 114, 82, 53, 31, 15, 7, 2, 1, 5, 13, 30, 56, 95, 150, 216, 293, 371, 445, 495, 522, 495, 445, 371, 293, 216, 150, 95, 56, 30, 13, 5, 1
Offset: 0
Examples
/\
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 ;
Links
- Alois P. Heinz, Rows n = 0..60, flattened
- Wikipedia, Counting lattice paths
Crossrefs
Programs
-
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 *)