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

%I A274405 #11 Dec 29 2020 09:04:01
%S A274405 0,0,0,1,6,34,179,915,4607,22988,114090,564359,2785921,13735074,
%T A274405 67665208,333211828,1640575047,8077199130,39770520844,195852723348,
%U A274405 964689515033,4752800817185,23422061819883,115456855588378,569293729146929,2807864888917275
%N A274405 Number of anti-down steps in all modified skew Dyck paths of semilength n.
%C A274405 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 A274405 Alois P. Heinz, <a href="/A274405/b274405.txt">Table of n, a(n) for n = 0..500</a>
%F A274405 a(n) = Sum_{k>0} k * A274404(n,k).
%F A274405 a(n) ~ c * 5^n / sqrt(n), where c = 0.0554525135364274199547478570703521322323... . - _Vaclav Kotesovec_, Jun 26 2016
%p A274405 b:= proc(x, y, t, n) option remember; `if`(y>n, 0, `if`(n=y,
%p A274405       `if`(t=2, 0, [1, 0]), b(x+1, y+1, 0, n-1)+`if`(t<>1
%p A274405        and x>0, (p-> p+[0, p[1]])(b(x-1, y+1, 2, n-1)), 0)+
%p A274405       `if`(t<>2 and y>0, b(x+1, y-1, 1, n-1), 0)))
%p A274405     end:
%p A274405 a:= n-> b(0$3, 2*n)[2]:
%p A274405 seq(a(n), n=0..30);
%t A274405 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]]];
%t A274405 a[n_] := b[0, 0, 0, 2 n][[2]];
%t A274405 a /@ Range[0, 30] (* _Jean-François Alcover_, Dec 29 2020, after _Alois P. Heinz_ *)
%Y A274405 Cf. A230823, A274404.
%K A274405 nonn
%O A274405 0,5
%A A274405 _Alois P. Heinz_, Jun 20 2016