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%I #45 Sep 08 2022 08:44:49
%S 1,2,1,1,3,6,14,33,79,194,482,1214,3090,7936,20544,53545,140399,
%T 370098,980226,2607242,6961462,18652112,50133616,135140598,365254226,
%U 989614976,2687312752,7312725944,19938170096,54460115308,149007155356,408341969073
%N a(n) = a(1)*a(n-1) + a(2)*a(n-2) + ... + a(n-3)*a(3) for n >= 4.
%C a(n) is the number of Dyck (n-1)-paths that contain no DUDU's and no UUDD's (n>=3). For example, a(5)=3 counts UUUDUDDD, UDUUDUDD, UUDUDDUD. - _David Callan_, Sep 25 2006
%C Apart from the first three terms, the total number of bargraphs of semiperimeter n with no levels for n>=2. The sequence begins: 1, 3, 6, 14, 33, ... - _Arnold Knopfmacher_, Nov 01 2016
%H Seiichi Manyama, <a href="/A025243/b025243.txt">Table of n, a(n) for n = 1..2192</a>
%H Paul Barry, <a href="https://arxiv.org/abs/1807.05794">Riordan Pseudo-Involutions, Continued Fractions and Somos 4 Sequences</a>, arXiv:1807.05794 [math.CO], 2018.
%H Paul Barry, <a href="https://arxiv.org/abs/1910.00875">Generalized Catalan recurrences, Riordan arrays, elliptic curves, and orthogonal polynomials</a>, arXiv:1910.00875 [math.CO], 2019.
%H Aubrey Blecher, Charlotte Brennan, Arnold Knopfmacher, <a href="https://doi.org/10.26493/1855-3974.600.5d2">Levels in bargraphs</a>, Ars Mathematica Contemporanea, 9 (2015), 287-300.
%F G.f.: (1 + x + 2*x^2 - sqrt(1 - 2*x - 3*x^2 + 4*x^4))/2. - _Michael Somos_, Jun 08 2000
%F Conjecture: n*(n+1)*a(n) +n*(n+1)*a(n-1) +(n^2+n+12)*a(n-2) +(-29*n^2+85*n+78)*a(n-3) +2*(-13*n^2+56*n+33)*a(n-4) +12*(n+1)*(n-7)*a(n-5) +8*(5*n+3)*(n-8)*a(n-6)=0. - _R. J. Mathar_, Feb 25 2015
%t CoefficientList[Series[(1 + x + 2 x^2 - Sqrt[1 - 2 x - 3 x^2 + 4 x^4]) / 2 / x, {x, 0, 40}], x] (* _Vincenzo Librandi_, Feb 15 2015 *)
%o (PARI) a(n)=polcoeff((x+2*x^2-sqrt(1-2*x-3*x^2+4*x^4+x*O(x^n)))/2,n)
%o (Magma) a:=[1,2,1]; for n in [4..35] do Append(~a,&+[a[k]*a[n-k]:k in [1..n-3]] ); end for; a; // _Marius A. Burtea_, Jan 02 2020
%K nonn
%O 1,2
%A _Clark Kimberling_
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