A274405 Number of anti-down steps in all modified skew Dyck paths of semilength n.

0, 0, 0, 1, 6, 34, 179, 915, 4607, 22988, 114090, 564359, 2785921, 13735074, 67665208, 333211828, 1640575047, 8077199130, 39770520844, 195852723348, 964689515033, 4752800817185, 23422061819883, 115456855588378, 569293729146929, 2807864888917275

Offset: 0

Author

Alois P. Heinz, Jun 20 2016

Keywords

• nonn

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

Crossrefs

Cf. A230823, A274404.

Programs

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 *)

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