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

1, 3, 10, 36, 136, 529, 2095, 8393, 33885, 137547, 560544, 2291181, 9386584, 38525224, 158350133, 651645511, 2684323326, 11066714500, 45656997415, 188475894929, 778444106703, 3216562337420, 13296099775859, 54979806370840, 227410731720624, 940875886301665

Offset: 0

Author

Florian Schager, Jan 23 2024

Keywords

• nonn
• walk

Comments

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.

A catastrophe is a step C = (1,-k) from altitude k to altitude 0 for k > 1.

Examples

			For n = 2 the a(2) = 10 solutions are UU, UR, UB, UD, RU, RR, RB, BU, BR, BB.
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.
		

Links

• Cyril Banderier and Michael Wallner, Lattice paths with catastrophes, arXiv:1707.01931 [math.CO], 2017.

Crossrefs

Cf. A274115 (Dyck meanders).

Cf. A054391 (Motzkin meanders).

Programs

Maple

K := 1 - z*(1/u + 2 + u);
u1 := solve(K, u)[2];
M := (1 - u1)/(1 - 4*z);
E := u1/z;
M1 := z*E^2;Q := z*(M - E - M1);
series(M/(1 - Q), z, 30);

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))

Formula

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).

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