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A369432 Number of Dyck excursions with catastrophes from (0,0) to (n,0).

%I A369432 #18 Jul 24 2025 10:51:55
%S A369432 1,1,3,6,16,37,95,230,582,1434,3606,8952,22446,55917,140007,349374,
%T A369432 874150,2183230,5460506,13643972,34118328,85270626,213205958,
%U A369432 532926716,1332420796,3330739972,8327221380,20816939100,52043684970,130105200765,325267849335,813155081070
%N A369432 Number of Dyck excursions with catastrophes from (0,0) to (n,0).
%C A369432 A Dyck excursion is a lattice path with steps U = (1,1) and D = (1,-1) that does not go below the x-axis and ends at the x-axis.
%C A369432 A catastrophe is a step C = (1,-k) from altitude k to altitude 0 for k >= 0.
%H A369432 Alois P. Heinz, <a href="/A369432/b369432.txt">Table of n, a(n) for n = 0..2514</a>
%H A369432 Cyril Banderier and Michael Wallner, <a href="https://arxiv.org/abs/1707.01931">Lattice paths with catastrophes</a>, arXiv:1707.01931 [math.CO], 2017, p.7.
%F A369432 G.f.: (1 - sqrt(1 - 4*z^2))*(2*z - 1)/(z^2*(6*z - 3 + sqrt(1 - 4*z^2))).
%F A369432 a(n) ~ 3/8*(5/2)^n.
%e A369432 For n = 3 the a(3) = 6 solutions are UUC, UDC, UCC, CUD, CUC, CCC.
%e A369432 For n = 4 the a(4) = 16 solutions are UUUC, UUDD, UUDC, UUCC, UDUD, UDUC, UDCC, UCUD, UCUC, UCCC, CUUC, CUDC, CUCC, CCUD, CCUC, CCCC.
%p A369432 u1 := solve(1 - z*(1/u + u), u)[2];
%p A369432 M := (1 - u1)/(1 - 2*z);
%p A369432 E := u1/z;
%p A369432 F := E/(-M*z + 1);
%p A369432 series(F, z, 33);
%p A369432 # second Maple program:
%p A369432 b:= proc(x, y) option remember; `if`(x=0, `if`(y=0, 1, 0),
%p A369432       b(x-1, 0)+`if`(y>0, b(x-1, y-1), 0)+b(x-1, y+1))
%p A369432     end:
%p A369432 a:= n-> b(n, 0):
%p A369432 seq(a(n), n=0..31);  # _Alois P. Heinz_, Jan 23 2024
%t A369432 b[x_, y_] := b[x, y] = If[x == 0, If[y == 0, 1, 0], b[x-1, 0] + If[y > 0, b[x-1, y-1], 0] + b[x-1, y+1]];
%t A369432 a[n_] := b[n, 0];
%t A369432 Table[a[n], {n, 0, 31}] (* _Jean-François Alcover_, Jul 24 2025, after _Alois P. Heinz_ *)
%o A369432 (PARI) my(N=44,z='z+O('z^N)); Vec((1 - sqrt(1 -4*z^2))*(2*z - 1)/(z^2*(6*z - 3 + sqrt(1 - 4*z^2))))
%Y A369432 Cf. A054341 (Dyck meanders with catastrophes).
%Y A369432 Cf. A224747 (different model of catastrophes).
%K A369432 nonn,walk
%O A369432 0,3
%A A369432 _Florian Schager_, Jan 23 2024