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A026569
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a(n) = T(n,n), T given by A026568. Also a(n) = number of integer strings s(0),...,s(n) counted by T, such that s(n)=0.
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8
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1, 1, 3, 5, 13, 27, 67, 153, 375, 893, 2189, 5319, 13089, 32155, 79479, 196573, 487833, 1212135, 3018355, 7525585, 18792303, 46980373, 117589689, 294613155, 738844719, 1854484305, 4658460165, 11710592711, 29458662005, 74151824271
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OFFSET
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0,3
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COMMENTS
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Number of grand Motzkin n-paths avoiding UF. - David Scambler, Jun 20 2013
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LINKS
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FORMULA
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a(n) = Sum_{k=0..floor(n/2)} binomial(2*k, k)*binomial(n-k, k). - Paul Barry, Sep 09 2004
G.f.: sqrt(1/((1-x)*(1-x-4*x^2))). - Ralf Stephan, Jan 08 2004
D-finite with recurrence: a(n) = 1/n*((2*n-1)*a(n-1) + (3*n-3)*a(n-2) - (4*n-6)*a(n-3)). - Vladeta Jovovic, Mar 12 2005
a(n) = Sum_{k=0..n} C(k, n-k)*C(2*(n-k), n-k). - Paul Barry, Jul 30 2005
G.f.: 1/(1-x-2*x^2/(1-0*x-x^2/(1-x-x^2/(1-0*x-2*x^2/(1-x-x^2/.... (continued fraction). Paul Barry, Dec 07 2008
a(n) ~ sqrt((5+13/sqrt(17))/8) * ((1+sqrt(17))/2)^n/sqrt(Pi*n). - Vaclav Kotesovec, Aug 10 2013
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EXAMPLE
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For a(3) = 5 the five grand Motzkin paths are FDU, DFU, FUD, UDF and FFF. The paths containing UF, namely UFD and DUF, are avoided. - David Scambler, Jun 20 2013
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MATHEMATICA
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CoefficientList[Series[Sqrt[1/((1-x)(1-x-4x^2))], {x, 0, 30}], x] (* Harvey P. Dale, Oct 06 2011 *)
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PROG
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(PARI) my(x='x+O('x^30)); Vec( 1/sqrt((1-x)*(1-x-4*x^2)) ) \\ G. C. Greubel, Aug 03 2019
(Magma) R<x>:=PowerSeriesRing(Rationals(), 30); Coefficients(R!( 1/Sqrt((1-x)*(1-x-4*x^2)) )); // G. C. Greubel, Aug 03 2019
(Sage) (1/sqrt((1-x)*(1-x-4*x^2))).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, Aug 03 2019
(GAP) List([0..30], n-> Sum([0..Int(n/2)], k-> Binomial(2*k, k)*Binomial( n-k, k) )); # G. C. Greubel, Aug 03 2019
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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