login
This site is supported by donations to The OEIS Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A026107 Second differences of Motzkin numbers (A001006). 5
1, 3, 7, 18, 46, 120, 316, 841, 2257, 6103, 16611, 45475, 125139, 345957, 960417, 2676291, 7483299, 20989833, 59042805, 166520124, 470781528, 1333970190, 3787707322, 10775741271, 30711538351, 87677551081, 250704001213, 717923179762 (list; graph; refs; listen; history; text; internal format)
OFFSET

2,2

COMMENTS

Number of (s(0), s(1), ..., s(n)) such that every s(i) is a nonnegative integer, s(0) = 0, s(1) = 1 = s(n), |s(i) - s(i-1)| <= 1 for i >= 2. Also a(n) = T(n,n-1), where T is array in A026105 and U(n,n+1), where U is array in A026120.

Also number of (s(0),s(1),...,s(n)) such that every s(i) is a nonnegative integer, s(0) = 1, s(n) = 0, |s(1) - s(0)| = 1, |s(i) - s(i-1)| <= 1 for i >= 2.

Number of Motzkin paths of length n+1 that start with a (1,1) step and end with a (1,-1) step. - Emeric Deutsch, Jul 11 2001

The sequence 1,1,3,7,18.... has a(n)=sum{k=0..n, C(n,2k)*A000108(k+1) }. - Paul Barry, Jul 18 2003

Equals iterates of M * [1,1,1,1,0,0,0,...] where M = an infinite tridiagonal matrix with [0,1,1,1,...] in the main diagonal and [1,1,1,...] in the super and subdiagonals. [From Gary W. Adamson, Jan 08 2009]

Motzkin paths of length n-1 that are allowed to go down to the line y=-1 [He-Shapiro, page 38]. - R. J. Mathar, Jul 23 2017

LINKS

Table of n, a(n) for n=2..29.

T.-X. He, L. W. Shapiro, Fuss-Catalan matrices, their weighted sums, and stabilizer subgroups of the Riordan group, Lin. Alg. Applic. 532 (2017) 25-41

FORMULA

a(n)=A001006(n+1)-2A001006(n)+A001006(n-1); g.f.: [(1-z)^2*M-1+z-z^2]/z, where M is the generating function of the Motzkin sequence A001006 (M = 1 + zM + z^2M^2).

(n+3)*a(n) +3*(-n-1)*a(n-1) +(-n-3)*a(n-2) +3*(n-3)*a(n-3)=0. - R. J. Mathar, Nov 26 2012

CROSSREFS

Cf. A001006. First differences of A002026.

Cf. A026122.

Sequence in context: A052960 A059512 A094297 * A173765 A027969 A027971

Adjacent sequences:  A026104 A026105 A026106 * A026108 A026109 A026110

KEYWORD

nonn

AUTHOR

Clark Kimberling

EXTENSIONS

Simpler definition from Ralf Stephan, Dec 16 2004

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent | More pages
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy .

Last modified February 24 12:46 EST 2018. Contains 299623 sequences. (Running on oeis4.)