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

0,3

Links

Alois P. Heinz, Rows n = 0..60, flattened

Wikipedia, Counting lattice paths

Formula

T(n,k) = T(n,-k).

T(n,A130518(n)) = A177702(n).

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

Row sums give A000108.

Column k=0 gives A300323.

Cf. A130518, A129182, A177702, A239927, A298645, A300953.

Sequence in context: A279945 A342724 A347046 * A144220 A156826 A130296

Adjacent sequences: A300319 A300320 A300321 * A300323 A300324 A300325

Keyword

nonn,tabf

Author

Alois P. Heinz, Mar 02 2018

Status

approved