A300322 -id:A300322 - cp's OEIS frontend

A300953 Number T(n,k) of Dyck paths of semilength n such that 2*k is the difference between the area above the path and the area below the path, measured within the smallest enclosing rectangle based on the x-axis; triangle T(n,k), n>=0, -floor((n-1)^2/4) <= k <= floor((n-1)^2/4), read by rows.

1, 1, 2, 1, 2, 2, 2, 0, 7, 0, 5, 1, 2, 3, 6, 7, 8, 6, 6, 3, 2, 0, 9, 0, 20, 0, 35, 0, 34, 0, 25, 0, 7, 1, 2, 4, 8, 10, 17, 23, 30, 38, 43, 46, 48, 42, 41, 26, 26, 12, 8, 4, 2, 0, 11, 0, 29, 0, 63, 0, 115, 0, 176, 0, 238, 0, 255, 0, 230, 0, 169, 0, 92, 0, 41, 0, 9

Offset: 0

Alois P. Heinz, Mar 16 2018

			              .______.
              | /\/\ |  ,  rectangle area: 12, above path area: 5,
T(3,-1) = 1:  |/____\|  ,  below path area: 7, difference: (5-7) = 2 * (-1).
.
                 /\
                /  \
T(3,0) = 2:    /    \   /\/\/\  .
.
                /\         /\
T(3,1) = 2:    /  \/\   /\/  \  .
.
Triangle T(n,k) begins:
:                                   1                                    ;
:                                   1                                    ;
:                                   2                                    ;
:                               1,  2,  2                                ;
:                           2,  0,  7,  0,  5                            ;
:                   1,  2,  3,  6,  7,  8,  6,  6,  3                    ;
:           2,  0,  9,  0, 20,  0, 35,  0, 34,  0, 25,  0,  7            ;
:  1, 2, 4, 8, 10, 17, 23, 30, 38, 43, 46, 48, 42, 41, 26, 26, 12, 8, 4  ;

Alois P. Heinz, Rows n = 0..50, flattened
Wikipedia, Counting lattice paths

Row sums give A000108.
Column k=0 gives A300952.
Cf. A002620, A026741, A040001, A129182, A239927, A300322, A300996.

Sum_{k = -floor((n-1)^2/4)..floor((n-1)^2/4)} k * T(n,k) = A300996(n).
T(n,-floor((n-1)^2/4)) = A040001(n).
T(n, floor((n-1)^2/4)) = A026741(n+1) for n > 2.
T(n,k) = 0 iff n is even and k is odd or abs(k) > floor(n*(n-1)/6).

A300323 Number of Dyck paths of semilength n such that the area under the right half of the path equals the area under the left half of the path.

Original entry on oeis.org

1, 1, 2, 3, 6, 12, 28, 69, 186, 522, 1536, 4638, 14408, 45568, 146884, 479871, 1589516, 5320854, 18000198, 61412376, 211282386, 731973720, 2553168136, 8957554412, 31604599044, 112060048354, 399227283950, 1428315878002, 5130964125124, 18499652813682

Offset: 0

Alois P. Heinz, Mar 02 2018

nonn

			              /\
             /  \      /\/\
a(3) = 3:   /    \    /    \    /\/\/\ .
.
a(5) = 12 counts A001405(5) = 10 symmetric plus 2 non-symmetric Dyck paths:
             /\  /\
          /\/  \/  \  and its reversal.

Alois P. Heinz, Table of n, a(n) for n = 0..300
Wikipedia, Counting lattice paths

Column k=0 of A300322.

Cf. A000108 (all Dyck paths), A000225, A001405 (symmetric Dyck paths), A129182, A239927, A298645.

b:= proc(x, y) option remember; expand(`if`(x=0, 1,
      `if`(y<1,   0, b(x-1, y-1)*z^(2*y-1))+
      `if`(x add(coeff(p, z, i)^2
      , i=0..degree(p)))(b(n, n-2*j)), j=0..n/2)
    end:
seq(a(n), n=0..32);

b[x_, y_] := b[x, y] = Expand[If[x == 0, 1, If[y < 1, 0, b[x - 1, y - 1] z^(2y - 1)] + If[x < y + 2, 0, b[x - 1, y + 1] z^(2y + 1)]]];
a[n_] := a[n] = Sum[Function[p, Sum[Coefficient[p, z, i]^2, {i, 0, Exponent[p, z]}]][b[n, n - 2j]], {j, 0, n/2}];
Table[a[n], {n, 0, 32}] (* Jean-François Alcover, May 31 2018, from Maple *)

a(n) >= A001405(n) with equality only for n <= 4.

a(n) is odd <=> n in { A000225 }.

cp's OEIS Frontend

A300953 Number T(n,k) of Dyck paths of semilength n such that 2*k is the difference between the area above the path and the area below the path, measured within the smallest enclosing rectangle based on the x-axis; triangle T(n,k), n>=0, -floor((n-1)^2/4) <= k <= floor((n-1)^2/4), read by rows.

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