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A230823 Number of modified skew Dyck paths of semilength n. 7
1, 1, 2, 6, 20, 73, 281, 1124, 4627, 19474, 83421, 362528, 1594389, 7083078, 31738724, 143281473, 651048571, 2975243348, 13665866849, 63055369522, 292130900461, 1358415528683, 6337824891559, 29660089051015, 139193062791189, 654903798282528, 3088627236146085 (list; graph; refs; listen; history; text; internal format)
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, Table of n, a(n) for n = 0..600

FORMULA

a(n) ~ c * 5^n / n^(3/2), where c = 0.27726256768213709977373928535... . - Vaclav Kotesovec, Jul 16 2014

G.f.: 1/(1 - x/(1 - (x + x^2)/(1 - (x + x^2 + x^3)/(1 - (x + x^2 + x^3 + x^4)/(1 - ...))))), a continued fraction (conjecture). - Ilya Gutkovskiy, Jun 08 2017

EXAMPLE

a(0) = 1: the empty path.

a(1) = 1: UD.

a(2) = 2: UUDD, UDUD.

a(3) = 6: UUUDDD, UUDUDD, UUDDUD, UAUDDD, UDUUDD, UDUDUD.

a(4) = 20: UUUUDDDD, UUUDUDDD, UUUDDUDD, UUUDDDUD, UUAUDDDD, UUDUUDDD, UUDUDUDD, UUDUDDUD, UUDDUUDD, UUDDUDUD, UAUUDDDD, UAUDUDDD, UAUDDUDD, UAUDDDUD, UDUUUDDD, UDUUDUDD, UDUUDDUD, UDUAUDDD, UDUDUUDD, UDUDUDUD.

a(5) = 73: UUUUUDDDDD, UUUUDUDDDD, UUUUDDUDDD, ..., UDUDUAUDDD, UDUDUDUUDD, UDUDUDUDUD.

MAPLE

b:= proc(x, y, t, n) option remember; `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), 0)+

      `if`(t<>2 and y>0, b(x+1, y-1, 1, n-1), 0)))

    end:

a:= n-> b(0$3, 2*n):

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, 1], b[x+1, y+1, 0, n-1] + If[t != 1 && x > 0, 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]; Table[a[n], {n, 0, 30}] (* Jean-Fran├žois Alcover, Dec 16 2013, translated from Maple *)

CROSSREFS

Cf. A000108, A128714.

Row sums of A274372 and of A274404.

Sequence in context: A052884 A150140 A061396 * A192497 A104632 A194956

Adjacent sequences:  A230820 A230821 A230822 * A230824 A230825 A230826

KEYWORD

nonn

AUTHOR

David Scambler and Alois P. Heinz, Oct 31 2013

STATUS

approved

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Last modified May 16 21:13 EDT 2022. Contains 353720 sequences. (Running on oeis4.)