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

0,5

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

Cf. A230823, A274404.

Sequence in context: A344717 A266359 A198765 * A144142 A126643 A320745

Adjacent sequences: A274402 A274403 A274404 * A274406 A274407 A274408

Keyword

nonn

Author

Alois P. Heinz, Jun 20 2016

Status

approved