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

Author

Clark Kimberling

Keywords

• nonn

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).

Crossrefs

First differences of A102071.

Programs

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 *)

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

Extensions

More terms from Ralf Stephan, Dec 30 2004