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A369436 Number of 2-Motzkin meanders with catastrophes of length n.

%I A369436 #15 May 01 2024 09:00:59
%S A369436 1,3,10,36,136,529,2095,8393,33885,137547,560544,2291181,9386584,
%T A369436 38525224,158350133,651645511,2684323326,11066714500,45656997415,
%U A369436 188475894929,778444106703,3216562337420,13296099775859,54979806370840,227410731720624,940875886301665
%N A369436 Number of 2-Motzkin meanders with catastrophes of length n.
%C A369436 A 2-Motzkin meander is a lattice path that does not go below the x-axis. with steps U = (1,1), D = (1,-1) and two copies R = (1,0) and B = (1,0), i.e. red and blue level steps.
%C A369436 A catastrophe is a step C = (1,-k) from altitude k to altitude 0 for k > 1.
%H A369436 Cyril Banderier and Michael Wallner, <a href="https://arxiv.org/abs/1707.01931">Lattice paths with catastrophes</a>, arXiv:1707.01931 [math.CO], 2017.
%F A369436 G.f.: (1 - 4*z - sqrt(1 - 4*z))*z/(5*sqrt(1 - 4*z)*z^2 + 12*z^3 - 5*z*sqrt(-4*z + 1) - 15*z^2 + sqrt(-4*z + 1) + 7*z - 1).
%F A369436 D-finite with recurrence n*a(n) +4*(-3*n+2)*a(n-1) +(53*n-70)*a(n-2) +(-107*n+210)*a(n-3) +(101*n-268)*a(n-4) +18*(-2*n+7)*a(n-5)=0. - _R. J. Mathar_, Jan 28 2024
%e A369436 For n = 2 the a(2) = 10 solutions are UU, UR, UB, UD, RU, RR, RB, BU, BR, BB.
%e A369436 For n = 3 the a(3) = 36 solutions are UUU, UUR, UUB, UUD, UUC, URU, URR, URB, URD, UBU, UBR, UBB, UBD, UDU, UDR, UDB, RUU, RUR, RUB, RUD, RRU, RRR, RRB, RBU, RBR, RBB, BUU, BUR, BUB, BUD, BRU, BRR, BRB, BBU, BBR, BBB.
%p A369436 K := 1 - z*(1/u + 2 + u);
%p A369436 u1 := solve(K, u)[2];
%p A369436 M := (1 - u1)/(1 - 4*z);
%p A369436 E := u1/z;
%p A369436 M1 := z*E^2;Q := z*(M - E - M1);
%p A369436 series(M/(1 - Q), z, 30);
%o A369436 (PARI) my(N=44,z='z+O('z^N),S=sqrt(1-4*z)); Vec((1 - 4*z - S)*z/(5*S*z^2 + 12*z^3 - 5*z*S - 15*z^2 + S + 7*z - 1))
%Y A369436 Cf. A274115 (Dyck meanders).
%Y A369436 Cf. A054391 (Motzkin meanders).
%K A369436 nonn,walk
%O A369436 0,2
%A A369436 _Florian Schager_, Jan 23 2024