A026269 a(n) = number of (s(0), s(1), ..., s(n)) such that every s(i) is a nonnegative integer, s(0) = 0 = s(n), s(1) = 1, |s(i) - s(i-1)| <= 1 for i >= 2, |s(2) - s(1)| = 1, |s(3) - s(2)| = 1 if s(2) = 1. Also a(n) = T(n,n) and a(n) = Sum{T(k,k-1)}, k = 1,2,...,n, where T is array in A026268.

1, 2, 4, 10, 25, 64, 166, 436, 1157, 3098, 8360, 22714, 62086, 170614, 471096, 1306374, 3636708, 10159590, 28473132, 80032638, 225562929, 637301652, 1804751718, 5121677512, 14563448593, 41487279622, 118389089432, 338381552294, 968627180975

Offset

2,2

Comments

Convolution of [1,2,3,6,13,..] (A005554) with [1,0,1,2,5,12...] (essentially A002026). - R. J. Mathar, Nov 01 2021

Links

Vincenzo Librandi, Table of n, a(n) for n = 2..1000

Gennady Eremin, Arithmetic on Balanced Parentheses: The case of Ordered Motzkin Words, arXiv:1911.01673 [math.CO], 2019. See (4.3) p. 13 (with a different offset).

Formula

G.f.: 4z^2(1-z^2)/[1-z+sqrt(1-2z-3z^2)]^2.

D-finite with recurrence (n+2)*a(n) +(-3*n-1)*a(n-1) +(-n+2)*a(n-2) +3*(n-5)*a(n-3)=0. - R. J. Mathar, Jun 10 2013

a(n) ~ 8 * 3^(n-3/2) / (sqrt(Pi) * n^(3/2)). - Vaclav Kotesovec, Feb 12 2014

a(n) = A002026(n-1) - A002026(n-3). - R. J. Mathar, Nov 01 2021

Mathematica

Drop[CoefficientList[Series[4x^2(1-x^2)/(1-x+Sqrt[1-2x-3x^2])^2, {x, 0, 30}], x], 2] (* Harvey P. Dale, May 05 2011 *)

Crossrefs

First differences of A102071.

Sequence in context: A036887 A307578 A151536 * A000645 A005958 A166516

Adjacent sequences: A026266 A026267 A026268 * A026270 A026271 A026272

Keyword

nonn

Author

Clark Kimberling

Extensions

More terms from Ralf Stephan, Dec 30 2004

Status

approved