A176677 Sequence defined by the recurrence formula a(n+1)=sum(a(p)*a(n-p)+k,p=0..n)+l for n>=1, with here a(0)=1, a(1)=1, k=0 and l=-1.

1, 1, 1, 2, 5, 14, 41, 123, 375, 1158, 3615, 11393, 36209, 115940, 373709, 1211740, 3949969, 12937612, 42558745, 140547051, 465799527, 1548766044, 5164917003, 17271369744, 57900615135, 194558333460, 655168354935, 2210681734671

Offset

0,4

Comments

a(n) is the number of Motzkin paths of length n-1 in which the (1,0)-steps at odd levels come in 2 colors. Example: a(5)=14 because, denoting U=(1,1), H=(1,0), and D=(1,-1), we have 2 paths of shape HUHD, 2 paths of shape UHDH, 4 paths of shape UHHD, and 1 path of each of the shapes HHHH, HHUD, HUDH, UDHH, UDUD, and UUDD. - Emeric Deutsch, May 02 2011

Links

Table of n, a(n) for n=0..27.

Petr Gregor, Torsten Mütze, and Namrata, Combinatorial generation via permutation languages. VI. Binary trees, arXiv:2306.08420 [cs.DM], 2023.

Formula

G.f.: (1-sqrt(1-4*z*(a(0)-z*a(0)^2+z*a(1)+(k+j)*z^2/(1-z)+k*z^2/(1-z)^2)))/(2*z) (k=0, j=-1).

G.f. G=G(z) satisfies the equation z^2*(1-z)*G^2-(1-z)*(1-2*z)*G+1-2*z =0. - Emeric Deutsch, May 02 2011

Conjecture: (n+1)*a(n) +2*(-3*n+1)*a(n-1) +(9*n-13)*a(n-2) -4*a(n-3) +4*(-n+4)*a(n-4)=0. - R. J. Mathar, Mar 01 2016

Example

a(2)=2*1*1-1=1. a(3)=2*1*1+1^1-1=2. a(4)=2*1*2+2*1*1-1=5.

Maple

l:=-1: : k := 0 :m :=1:d(0):=1:d(1):=m: for n from 1 to 30 do d(n+1):=sum(d(p)*d(n-p)+k, p=0..n)+l:od :
taylor((1-sqrt(1-4*z*(d(0)-z*d(0)^2+z*m+(k+l)*z^2/(1-z)+k*z^2/(1-z)^2)))/(2*z), z=0, 30); seq(d(n), n=0..30);

Crossrefs

Cf. A176675.

Sequence in context: A326254 A054391 A365508 * A365510 A108626 A365509

Adjacent sequences: A176674 A176675 A176676 * A176678 A176679 A176680

Keyword

nonn

Author

Richard Choulet, Apr 23 2010

Status

approved