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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.

%I A026269 #24 Oct 28 2022 17:10:43
%S A026269 1,2,4,10,25,64,166,436,1157,3098,8360,22714,62086,170614,471096,
%T A026269 1306374,3636708,10159590,28473132,80032638,225562929,637301652,
%U A026269 1804751718,5121677512,14563448593,41487279622,118389089432,338381552294,968627180975
%N 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.
%C A026269 Convolution of [1,2,3,6,13,..] (A005554) with [1,0,1,2,5,12...] (essentially A002026). - _R. J. Mathar_, Nov 01 2021
%H A026269 Vincenzo Librandi, <a href="/A026269/b026269.txt">Table of n, a(n) for n = 2..1000</a>
%H A026269 Gennady Eremin, <a href="https://arxiv.org/abs/1911.01673">Arithmetic on Balanced Parentheses: The case of Ordered Motzkin Words</a>, arXiv:1911.01673 [math.CO], 2019. See (4.3) p. 13 (with a different offset).
%F A026269 G.f.: 4z^2(1-z^2)/[1-z+sqrt(1-2z-3z^2)]^2.
%F A026269 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
%F A026269 a(n) ~ 8 * 3^(n-3/2) / (sqrt(Pi) * n^(3/2)). - _Vaclav Kotesovec_, Feb 12 2014
%F A026269 a(n) = A002026(n-1) - A002026(n-3). - _R. J. Mathar_, Nov 01 2021
%t A026269 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 *)
%Y A026269 First differences of A102071.
%K A026269 nonn
%O A026269 2,2
%A A026269 _Clark Kimberling_
%E A026269 More terms from _Ralf Stephan_, Dec 30 2004