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 A026945 A bisection of the Motzkin numbers A001006. 7

%I

%S 1,2,9,51,323,2188,15511,113634,853467,6536382,50852019,400763223,

%T 3192727797,25669818476,208023278209,1697385471211,13933569346707,

%U 114988706524270,953467954114363,7939655757745265,66368199913921497

%N A bisection of the Motzkin numbers A001006.

%C a(n) = sum of the squares of numbers in row n of array T given by A026300.

%C Number of closed walks of length 2n on the one-way infinite ladder graph starting from (and ending at) a node of degree 2. - Mitch Harris, Mar 06 2004

%C a(n) = number of ways to connect 2n points labeled 1,2,...,2n in a line with 0 or more noncrossing arcs. For example, with arcs separated by dashes, a(2)=9 counts {} (no arcs), 12, 13, 14, 23, 24, 34, 12-34, 14-23. - _David Callan_, Sep 18 2007

%D Veronika Irvine, Lace Tessellations: A mathematical model for bobbin lace and an exhaustive combinatorial search for patterns, PhD Dissertation, University of Victoria, 2016.

%H Vincenzo Librandi, <a href="/A026945/b026945.txt">Table of n, a(n) for n = 0..200</a>

%H I. Dolinka, J. East, A. Evangelou, D. FitzGerald, N. Ham, <a href="http://arxiv.org/abs/1507.04838">Idempotent Statistics of the Motzkin and Jones Monoids</a>, arXiv preprint arXiv:1507.04838, 2015

%F a(n) = A005043(2n) + A005043(2n+1). - _Ralf Stephan_, Feb 06 2004

%F a(n) = sum{k=0..n, binomial(2n,2k)*C(k)}, C(n)=A000108(n); - _Paul Barry_, Jul 11 2008

%F a(n) = (2/Pi)*integral(x=-1..1, (1+2*x)^(2*n)*sqrt(1-x^2)). - _Peter Luschny_, Sep 11 2011

%F Recurrence: (n+1)*(2*n+1)*a(n) = (14*n^2+9*n-2)*a(n-1) + 3*(14*n^2-51*n+43)*a(n-2) - 27*(n-2)*(2*n-5)*a(n-3). - _Vaclav Kotesovec_, Oct 08 2012

%F a(n) ~ 3^(2*n+3/2)/(2^(5/2)*sqrt(Pi)*n^(3/2)). - _Vaclav Kotesovec_, Oct 08 2012

%F G.f.: (1/x) * Series_Reversion( x * (1-x) * (1-2*x)^2 / (1 - 3*x + 3*x^2) ). - _Paul D. Hanna_, Oct 03 2014

%F From _Peter Luschny_, May 15 2016: (Start)

%F a(n) = ((9-9*n)*(2*n-3)*(4*n+1)*a(n-2)+((8*n-2))*(10*n^2-5*n-3)*a(n-1))/((1+2*n)*(4*n-3)*(n+1)) for n>=2.

%F a(n) = hypergeom([1/2-n, -n], [2], 4). (End)

%p G:=(1-x-(1-2*x-3*x^2)^(1/2))/(2*x^2): GG:=series(G,x=0,60): 1, seq(coeff(GG,x^(2*n)),n=1..23);

%p a := n -> hypergeom([1/2-n, -n], [2], 4);

%p seq(simplify(a(n)), n=0..29); # _Peter Luschny_, May 15 2016

%t Table[SeriesCoefficient[(1-x-Sqrt[1-2*x-3*x^2])/(2*x^2),{x,0,2*n}],{n,0,20}] (* _Vaclav Kotesovec_, Oct 08 2012 *)

%o (PARI)

%o C(n)=binomial(2*n,n)/(n+1);

%o a(n)=sum(k=0,n, binomial(2*n,2*k)*C(k));

%o \\ _Joerg Arndt_, May 04 2013

%o (PARI) {a(n)=polcoeff(1/x*serreverse( x * (1-x) * (1-2*x)^2 /(1 - 3*x + 3*x^2 +x^2*O(x^n)) ),n)}

%o for(n=0,30,print1(a(n),", ")) \\ _Paul D. Hanna_, Oct 03 2014

%Y Cf. A001006, A099250.

%K nonn,easy

%O 0,2

%A _Clark Kimberling_

%E Entry revised by _N. J. A. Sloane_, Nov 16 2004

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Last modified January 17 17:12 EST 2019. Contains 319235 sequences. (Running on oeis4.)