OFFSET
0,3
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..160, flattened
FORMULA
Sum_{k>0} k * T(n,k) = A274405(n).
EXAMPLE
/\
\ \
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;
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 *)
KEYWORD
nonn,tabf
AUTHOR
Alois P. Heinz, Jun 20 2016
STATUS
approved