OFFSET
0,5
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, Table of n, a(n) for n = 0..500
FORMULA
a(n) = Sum_{k>0} k * A274404(n,k).
a(n) ~ c * 5^n / sqrt(n), where c = 0.0554525135364274199547478570703521322323... . - Vaclav Kotesovec, Jun 26 2016
MAPLE
b:= proc(x, y, t, n) option remember; `if`(y>n, 0, `if`(n=y,
`if`(t=2, 0, [1, 0]), b(x+1, y+1, 0, n-1)+`if`(t<>1
and x>0, (p-> p+[0, p[1]])(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:
a:= n-> b(0$3, 2*n)[2]:
seq(a(n), n=0..30);
MATHEMATICA
b[x_, y_, t_, n_] := b[x, y, t, n] = If[y > n, 0, If[n == y, If[t == 2, {0, 0}, {1, 0}], b[x + 1, y + 1, 0, n - 1] + If[t != 1 && x > 0, Function[p, p + {0, p[[1]]}][b[x - 1, y + 1, 2, n - 1]], 0] + If[t != 2 && y > 0, b[x + 1, y - 1, 1, n - 1], 0]]];
a[n_] := b[0, 0, 0, 2 n][[2]];
a /@ Range[0, 30] (* Jean-François Alcover, Dec 29 2020, after Alois P. Heinz *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Alois P. Heinz, Jun 20 2016
STATUS
approved