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A155867
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A 'Morgan Voyce' transform of the large Schroeder numbers A006318.
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1
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1, 3, 13, 65, 355, 2061, 12501, 78323, 503033, 3294373, 21916883, 147708777, 1006330457, 6919474163, 47956087733, 334658965641, 2349535729811, 16583609673797, 117608812053277, 837626242775875, 5988634758319665
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OFFSET
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0,2
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COMMENTS
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Image of A006318 under the Riordan array (1/(1-x), x/(1-x)^2).
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LINKS
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FORMULA
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G.f.: (1 - 3*x + x^2 - sqrt(1 - 10*x + 19*x^2 - 10*x^3 + x^4))/(2*x*(1-x)).
G.f.: 1/(1 -x -2*x/(1 -x -x/(1 -x -2*x/(1 -x -x/(1 -x -2*x/(1 -x -x/(1 - ... (continued fraction).
a(n) = Sum_{k=0..n} binomial(n+k,2k)*A006318(k).
Conjecture: (n+1)*a(n) + (4-11*n)*a(n-1) + (29*n-43)*a(n-2) +(73-29*n)*a(n-3) + (11*n-40)*a(n-4) + (5-n)*a(n-5) = 0. - R. J. Mathar, Jul 24 2012
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MATHEMATICA
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A006318[n_]:= 2*Hypergeometric2F1[-n+1, n+2, 2, -1];
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PROG
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(Magma) R<x>:=PowerSeriesRing(Rationals(), 40); Coefficients(R!( (1-3*x+x^2 -Sqrt(1-10*x+19*x^2-10*x^3+x^4))/(2*x*(1-x)) )); // G. C. Greubel, Jun 09 2021
(Sage)
P.<x> = PowerSeriesRing(ZZ, prec)
return P( (1-3*x+x^2 -sqrt(1-10*x+19*x^2-10*x^3+x^4))/(2*x*(1-x)) ).list()
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CROSSREFS
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KEYWORD
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easy,nonn
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AUTHOR
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STATUS
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approved
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