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A230823 Number of modified skew Dyck paths of semilength n.

%I A230823 #29 Jun 16 2022 10:37:35
%S A230823 1,1,2,6,20,73,281,1124,4627,19474,83421,362528,1594389,7083078,
%T A230823 31738724,143281473,651048571,2975243348,13665866849,63055369522,
%U A230823 292130900461,1358415528683,6337824891559,29660089051015,139193062791189,654903798282528,3088627236146085
%N A230823 Number of modified skew Dyck paths of semilength n.
%C A230823 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.
%H A230823 Alois P. Heinz, <a href="/A230823/b230823.txt">Table of n, a(n) for n = 0..600</a>
%F A230823 a(n) ~ c * 5^n / n^(3/2), where c = 0.27726256768213709977373928535... . - _Vaclav Kotesovec_, Jul 16 2014
%F A230823 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
%e A230823 a(0) = 1: the empty path.
%e A230823 a(1) = 1: UD.
%e A230823 a(2) = 2: UUDD, UDUD.
%e A230823 a(3) = 6: UUUDDD, UUDUDD, UUDDUD, UAUDDD, UDUUDD, UDUDUD.
%e A230823 a(4) = 20: UUUUDDDD, UUUDUDDD, UUUDDUDD, UUUDDDUD, UUAUDDDD, UUDUUDDD, UUDUDUDD, UUDUDDUD, UUDDUUDD, UUDDUDUD, UAUUDDDD, UAUDUDDD, UAUDDUDD, UAUDDDUD, UDUUUDDD, UDUUDUDD, UDUUDDUD, UDUAUDDD, UDUDUUDD, UDUDUDUD.
%e A230823 a(5) = 73: UUUUUDDDDD, UUUUDUDDDD, UUUUDDUDDD, ..., UDUDUAUDDD, UDUDUDUUDD, UDUDUDUDUD.
%p A230823 b:= proc(x, y, t, n) option remember; `if`(y>n, 0,
%p A230823       `if`(n=y, `if`(t=2, 0, 1), b(x+1, y+1, 0, n-1)+
%p A230823       `if`(t<>1 and x>0, b(x-1, y+1, 2, n-1), 0)+
%p A230823       `if`(t<>2 and y>0, b(x+1, y-1, 1, n-1), 0)))
%p A230823     end:
%p A230823 a:= n-> b(0$3, 2*n):
%p A230823 seq(a(n), n=0..30);
%t A230823 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-François Alcover_, Dec 16 2013, translated from Maple *)
%Y A230823 Cf. A000108, A128714.
%Y A230823 Row sums of A274372 and of A274404.
%K A230823 nonn
%O A230823 0,3
%A A230823 _David Scambler_ and _Alois P. Heinz_, Oct 31 2013