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%I #38 Nov 14 2019 05:35:12
%S 1,4,12,34,98,294,919,2974,9891,33604,116103,406614,1440025,5147876,
%T 18550572,67310938,245716094,901759950,3325066996,12312494462,
%U 45766188948,170702447074,638698318850,2396598337950
%N Partial sums of A014138.
%C Self-convolution of A014137. Column in triangle A200965. - _Philippe Deléham_, Jan 24 2014
%C For n >= 2, a(n-2) is the number of 021-avoiding ascent sequences of length n with exactly one occurrence of the consecutive pattern 01. For example, with n=3, a(1)=4 counts 001, 010, 011, 012. - _David Callan_, Nov 13 2019
%D Silvia Heubach and Toufik Mansour, Combinatorics of Compositions and Words, CRC Press, 2010.
%H Vincenzo Librandi, <a href="/A014143/b014143.txt">Table of n, a(n) for n = 0..200</a>
%H S. Kitaev, J. Remmel and M. Tiefenbruck, <a href="http://arxiv.org/abs/1201.6243">Marked mesh patterns in 132-avoiding permutations I</a>, arXiv preprint arXiv:1201.6243 [math.CO], 2012. - From _N. J. A. Sloane_, May 09 2012 [An early version on the arXiv had A014043 instead of A014143]
%H Sergey Kitaev, Jeffrey Remmel, Mark Tiefenbruck, <a href="http://www.emis.de/journals/INTEGERS/papers/p16/p16.Abstract.html">Quadrant Marked Mesh Patterns in 132-Avoiding Permutations II</a>, Electronic Journal of Combinatorial Number Theory, Volume 15 #A16. (<a href="http://arxiv.org/abs/1302.2274">arXiv</a>, arXiv:1302.2274 [math.CO], 2013)
%F G.f.: (1-2*z-sqrt(1-4*z))/(2*z^2*(1-z)^2). - _Emeric Deutsch_, Jan 27 2003
%F Recurrence: (n+2)*a(n) = 6*(n+1)*a(n-1) - 3*(3*n+2)*a(n-2) + 2*(2*n+1)*a(n-3). - _Vaclav Kotesovec_, Oct 07 2012
%F a(n) ~ 2^(2n+6)/(9*sqrt(Pi)*n^(3/2)). - _Vaclav Kotesovec_, Oct 07 2012
%F a(n) = 2 * Sum_{k=0..n} Sum_{j=0..k} C(2*j+1,j)/(j+2). - _Vaclav Kotesovec_, Oct 27 2012
%t Table[SeriesCoefficient[(1-2*x-Sqrt[1-4*x])/(2*x^2*(1-x)^2),{x,0,n}],{n,0,20}] (* _Vaclav Kotesovec_, Oct 07 2012 *)
%t Table[2*Sum[Sum[Binomial[2*j+1,j]/(j+2),{j,0,k}],{k,0,n}],{n,0,20}] (* _Vaclav Kotesovec_, Oct 27 2012 *)
%o (PARI) x='x+O('x^66); Vec((1-2*x-sqrt(1-4*x))/(2*x^2*(1-x)^2)) \\ _Joerg Arndt_, May 04 2013
%Y Cf. A014137, A200965.
%K nonn
%O 0,2
%A _N. J. A. Sloane_
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