A274372 Number T(n,k) of modified skew Dyck paths of semilength n such that the area between the x-axis and the path is k; triangle T(n,k), n>=0, n<=k<=n^2, read by rows.

1, 1, 1, 0, 1, 1, 0, 2, 1, 1, 0, 1, 1, 0, 3, 2, 3, 1, 3, 2, 2, 1, 1, 0, 1, 1, 0, 4, 3, 7, 4, 7, 5, 8, 6, 6, 3, 5, 4, 3, 2, 2, 1, 1, 0, 1, 1, 0, 5, 4, 12, 10, 17, 12, 20, 18, 22, 14, 19, 16, 18, 14, 14, 12, 12, 7, 8, 7, 5, 4, 3, 2, 2, 1, 1, 0, 1

Offset: 0

Author

Alois P. Heinz, Jun 19 2016

Keywords

• nonn
• tabf

Comments

A modified skew Dyck path is a path in the first quadrant which begins at the origin, ends on the x-axis, consists of steps U=(1,1) (up), D=(1,-1) (down) and A=(-1,1) (anti-down) so that A and D steps do not overlap.

Examples

			T(3,3) = 1:   /\/\/\
.
                 /\         /\
T(3,5) = 2:   /\/  \   ,   /  \/\
.
              /\
              \ \
T(3,6) = 1:   /  \
.
               /\/\
T(3,7) = 1:   /    \
.
                /\
               /  \
T(3,9) = 1:   /    \
.
Triangle T(n,k) begins:
n\k: 0 1 2 3 4 5 6 7 8 9 . . . . . .16 . . . . . . . .25
---+----------------------------------------------------
00 : 1
01 :   1
02 :     1 0 1
03 :       1 0 2 1 1 0 1
04 :         1 0 3 2 3 1 3 2 2 1 1 0 1
05 :           1 0 4 3 7 4 7 5 8 6 6 3 5 4 3 2 2 1 1 0 1
		

Links

• Alois P. Heinz, Rows n = 0..40, flattened

Crossrefs

Row sums give: A230823.

Column sums give: A274376.

Cf. A000217, A002061 (number of terms in row n), A129172, A274054, A274373.

Programs

Maple

b:= proc(x, y, t, n) option remember; expand(`if`(y>n, 0,
      `if`(n=y, `if`(t=2, 0, 1), b(x+1, y+1, 0, n-1)*z^
      (2*y+1)+`if`(t<>1 and x>0, b(x-1, y+1, 2, n-1), 0)+
      `if`(t<>2 and y>0, b(x+1, y-1, 1, n-1), 0))))
    end:
T:= n-> (p-> seq(coeff(p, z, i), i=n..n^2))(b(0$3, 2*n)):
seq(T(n), n=0..8);

Mathematica

b[x_, y_, t_, n_] := b[x, y, t, n] = Expand[If[y>n, 0, If[n == y, If[t == 2, 0, 1], b[x+1, y+1, 0, n-1]*z^(2*y+1) + If[t != 1 && x>0, b[x-1, y+1, 2, n-1], 0] + If[t != 2 && y>0, b[x+1, y-1, 1, n-1], 0]]]]; T[n_] := Function[p, Table[Coefficient[p, z, i], {i, n, n^2}]][b[0, 0, 0, 2*n]]; Table[T[n], {n, 0, 8}] // Flatten (* Jean-François Alcover, Jan 25 2017, translated from Maple *)

Formula

Sum_{k=n..n^2} k * T(n,k) = A274373(n).

T(n,n) = T(n,n^2) = 1.

T(n,n+1) = T(n,n^2-1) = 0.

T(n,n*(n+1)/2) = T(n,A000217(n)) = A274054(n).