cp's OEIS Frontend

A369982 Number of Dyck bridges with resets from any height to zero from (0,0) to (n,0).

%I A369982 #21 Feb 22 2025 10:43:24
%S A369982 1,1,5,11,39,105,335,965,2965,8755,26517,79047,238065,712347,2140473,
%T A369982 6414555,19256535,57743865,173280215,519743405,1559414971,4677875401,
%U A369982 14034331635,42101584041,126307456279,378916960525,1136761282175,3410263045325,10230829252575
%N A369982 Number of Dyck bridges with resets from any height to zero from (0,0) to (n,0).
%C A369982 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 A369982 A reset to zero is a step R = (1,-h) at altitude h for any integer h.
%H A369982 Alois P. Heinz, <a href="/A369982/b369982.txt">Table of n, a(n) for n = 0..1000</a>
%F A369982 G.f.: (2*z-1)/((3*z-1)*sqrt(1-4*z^2)).
%F A369982 a(n) ~ 3^n/sqrt(5).
%e A369982 For n = 3 the a(3) = 11 solutions are UUR, UDR, URR, DUR, DDR, DRR, RUD, RUR, RDU, RDR, RRR.
%p A369982 K := 1 - z*(u + 1/u);
%p A369982 v1, u1 := solve(K, u);
%p A369982 B := -z*diff(v1, z)/v1;
%p A369982 W := 1/(1 - 2*z);
%p A369982 series(B/(-W*z + 1), z, 30);
%p A369982 # second Maple program:
%p A369982 b:= proc(x, y) option remember; `if`(x=0, `if`(y=0, 1, 0),
%p A369982       b(x-1, 0)+b(x-1, abs(y-1))+b(x-1, y+1))
%p A369982     end:
%p A369982 a:= n-> b(n, 0):
%p A369982 seq(a(n), n=0..32);  # _Alois P. Heinz_, Feb 07 2024
%t A369982 b[x_, y_] := b[x, y] = If[x == 0, If[y == 0, 1, 0],
%t A369982    b[x - 1, 0] + b[x - 1, Abs[y - 1]] + b[x - 1, y + 1]];
%t A369982 a[n_] := b[n, 0];
%t A369982 Table[a[n], {n, 0, 32}] (* _Jean-François Alcover_, Feb 22 2025, after _Alois P. Heinz_ *)
%Y A369982 Cf. A369316 (for a different model of resets to zero).
%K A369982 nonn,walk
%O A369982 0,3
%A A369982 _Florian Schager_, Feb 07 2024