A274404 Number T(n,k) of modified skew Dyck paths of semilength n with exactly k anti-down steps; triangle T(n,k), n>=0, 0<=k<=n-floor((1+sqrt(max(0,8n-7)))/2), read by rows.

1, 1, 2, 5, 1, 14, 6, 42, 28, 3, 132, 120, 28, 1, 429, 495, 180, 20, 1430, 2002, 990, 195, 10, 4862, 8008, 5005, 1430, 165, 4, 16796, 31824, 24024, 9009, 1650, 117, 1, 58786, 125970, 111384, 51688, 13013, 1617, 70, 208012, 497420, 503880, 278460, 89180, 16016, 1386, 35

Offset: 0

Author

Alois P. Heinz, Jun 20 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(3,1) = 1:   /  \
.
Triangle T(n,k) begins:
:     1;
:     1;
:     2;
:     5,     1;
:    14,     6;
:    42,    28,     3;
:   132,   120,    28,    1;
:   429,   495,   180,   20;
:  1430,  2002,   990,  195,   10;
:  4862,  8008,  5005, 1430,  165,   4;
: 16796, 31824, 24024, 9009, 1650, 117, 1;
		

Links

• Alois P. Heinz, Rows n = 0..160, flattened

Crossrefs

Columns k=0-3 give: A000108, A002694(n-1), A074922(n-2), A232224(n-3).

Row sums give A230823.

Last elements of rows give A092392(n-1) for n>0.

Cf. A083920, A274405.

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)+
      `if`(t<>1 and x>0, b(x-1, y+1, 2, n-1)*z, 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=0..degree(p)))(b(0$3, 2*n)):
seq(T(n), n=0..14);

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] +
     If[t != 1 && x > 0, b[x - 1, y + 1, 2, n - 1] z, 0] +
     If[t != 2 && y > 0, b[x + 1, y - 1, 1, n - 1], 0]]]];
T[n_] := CoefficientList[b[0, 0, 0, 2n], z];
T /@ Range[0, 14] // Flatten (* Jean-François Alcover, Mar 27 2021, after Alois P. Heinz *)

Formula

Sum_{k>0} k * T(n,k) = A274405(n).