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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. 4
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 (list; graph; refs; listen; history; text; internal format)
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
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
KEYWORD
nonn
AUTHOR
EXTENSIONS
More terms from Ralf Stephan, Dec 30 2004
STATUS
approved

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Last modified July 15 12:56 EDT 2024. Contains 374332 sequences. (Running on oeis4.)