A026107 Second differences of Motzkin numbers (A001006).

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

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

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. - Gary W. Adamson, Jan 08 2009

Number of 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

With offset 1, a[n] = [x^n](1 + x + x^2)^n - [x^(n-4)](1 + x + x^2)^n, that is, the difference between the n-th central trinomial coefficient and its fourth predecessor. For example, with n = 4, (1 + x + x^2)^4 = 1 + 4*x + 10*x^2 + 16*x^3 + 19*x^4 + 16*x^5 + 10*x^6 + 4*x^7 + x^8 and a(4) = 19 - 1. - David Callan, Dec 18 2021

Links

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

T.-X. He and 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) - 2*A001006(n) + A001006(n-1).

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

G.f.: ((1-z)^2*M - 1 + z - z^2)/z, where M is the generating function of the Motzkin sequence A001006 (M = 1 + z*M + z^2*M^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

a(n) ~ 2 * 3^(n + 1/2) / (sqrt(Pi) * n^(3/2)). - Vaclav Kotesovec, Sep 17 2019

With offset 0 and a(0) = 1 prepended (see Paul Barry's formula above), a(n) = hypergeom([3/2, (1 - n)/2, -n/2], [1/2, 3], 4). - Peter Luschny, Dec 19 2021

Crossrefs

Cf. A001006. First differences of A002026.

Cf. A026122.

Sequence in context: A052960 A059512 A094297 * A372033 A173765 A027969

Adjacent sequences: A026104 A026105 A026106 * A026108 A026109 A026110

Keyword

nonn

Author

Clark Kimberling

Extensions

Simpler definition from Ralf Stephan, Dec 16 2004

Status

approved