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A026945
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A bisection of the Motzkin numbers A001006.
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7
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1, 2, 9, 51, 323, 2188, 15511, 113634, 853467, 6536382, 50852019, 400763223, 3192727797, 25669818476, 208023278209, 1697385471211, 13933569346707, 114988706524270, 953467954114363, 7939655757745265, 66368199913921497
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OFFSET
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0,2
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COMMENTS
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a(n) = sum of the squares of numbers in row n of array T given by A026300.
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
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
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REFERENCES
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Veronika Irvine, Lace Tessellations: A mathematical model for bobbin lace and an exhaustive combinatorial search for patterns, PhD Dissertation, University of Victoria, 2016.
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LINKS
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Vincenzo Librandi, Table of n, a(n) for n = 0..200
I. Dolinka, J. East, A. Evangelou, D. FitzGerald, N. Ham, Idempotent Statistics of the Motzkin and Jones Monoids, arXiv preprint arXiv:1507.04838, 2015.
Michael Torpey, Semigroup congruences: computational techniques and theoretical applications, Ph.D. Thesis, University of St. Andrews (Scotland, 2019).
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FORMULA
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a(n) = A005043(2n) + A005043(2n+1). - Ralf Stephan, Feb 06 2004
a(n) = sum{k=0..n, binomial(2n,2k)*C(k)}, C(n)=A000108(n); - Paul Barry, Jul 11 2008
a(n) = (2/Pi)*integral(x=-1..1, (1+2*x)^(2*n)*sqrt(1-x^2)). - Peter Luschny, Sep 11 2011
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
a(n) ~ 3^(2*n+3/2)/(2^(5/2)*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Oct 08 2012
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
From Peter Luschny, May 15 2016: (Start)
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.
a(n) = hypergeom([1/2-n, -n], [2], 4). (End)
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MAPLE
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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);
a := n -> hypergeom([1/2-n, -n], [2], 4);
seq(simplify(a(n)), n=0..29); # Peter Luschny, May 15 2016
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MATHEMATICA
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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 *)
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PROG
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(PARI)
C(n)=binomial(2*n, n)/(n+1);
a(n)=sum(k=0, n, binomial(2*n, 2*k)*C(k));
\\ Joerg Arndt, May 04 2013
(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)}
for(n=0, 30, print1(a(n), ", ")) \\ Paul D. Hanna, Oct 03 2014
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CROSSREFS
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Cf. A001006, A099250.
Sequence in context: A020087 A277378 A079836 * A246464 A009310 A091319
Adjacent sequences: A026942 A026943 A026944 * A026946 A026947 A026948
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KEYWORD
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nonn,easy
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AUTHOR
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Clark Kimberling
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EXTENSIONS
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Entry revised by N. J. A. Sloane, Nov 16 2004
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STATUS
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approved
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