A274405 -id:A274405 - cp's OEIS frontend

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

Alois P. Heinz, Jun 20 2016

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.

			              /\
              \ \
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;

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

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.

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);

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 *)

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

cp's OEIS Frontend

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.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula