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

0,8

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.

Links

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

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).

Example

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

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 *)

Crossrefs

Row sums give: A230823.

Column sums give: A274376.

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

Sequence in context: A261769 A005590 A142598 * A037800 A144411 A138253

Adjacent sequences: A274369 A274370 A274371 * A274373 A274374 A274375

Keyword

nonn,tabf

Author

Alois P. Heinz, Jun 19 2016

Status

approved