|
%I #17 Feb 14 2024 18:27:39
%S 1,1,5,11,39,105,335,965,2965,8755,26517,79047,238065,712347,2140473,
%T 6414555,19256535,57743865,173280215,519743405,1559414971,4677875401,
%U 14034331635,42101584041,126307456279,378916960525,1136761282175,3410263045325,10230829252575
%N Number of Dyck bridges with resets from any height to zero from (0,0) to (n,0).
%C A Dyck bridge is a lattice path with steps U = (1,1) and D = (1,-1) that is allowed to go below the x-axis and ends at altitude 0.
%C A reset to zero is a step R = (1,-h) at altitude h for any integer h.
%H Alois P. Heinz, <a href="/A369982/b369982.txt">Table of n, a(n) for n = 0..1000</a>
%F G.f.: (2*z-1)/((3*z-1)*sqrt(1-4*z^2)).
%F a(n) ~ 3^n/sqrt(5).
%e For n = 3 the a(3) = 11 solutions are UUR, UDR, URR, DUR, DDR, DRR, RUD, RUR, RDU, RDR, RRR.
%p K := 1 - z*(u + 1/u);
%p v1, u1 := solve(K, u);
%p B := -z*diff(v1, z)/v1;
%p W := 1/(1 - 2*z);
%p series(B/(-W*z + 1), z, 30);
%p # second Maple program:
%p b:= proc(x, y) option remember; `if`(x=0, `if`(y=0, 1, 0),
%p b(x-1, 0)+b(x-1, abs(y-1))+b(x-1, y+1))
%p end:
%p a:= n-> b(n, 0):
%p seq(a(n), n=0..32); # _Alois P. Heinz_, Feb 07 2024
%Y Cf. A369316 (for a different model of resets to zero).
%K nonn,walk
%O 0,3
%A _Florian Schager_, Feb 07 2024
|
