A369436 - OEIS

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,2

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.

LINKS

Table of n, a(n) for n=0..25.

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

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

EXAMPLE

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.

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);

PROG

(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))

CROSSREFS

Cf. A274115 (Dyck meanders).

Cf. A054391 (Motzkin meanders).