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A243585
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Expansion of x*log'(C(C(x)-1)-1), C(x) = (1-sqrt(1-4*x))/(2*x).
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2
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1, 4, 20, 106, 580, 3244, 18446, 106250, 618340, 3628600, 21438820, 127377980, 760346350, 4556473276, 27396081950, 165189725326, 998492094244, 6048338850560, 36706629690824, 223139239595840, 1358475322091620
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OFFSET
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0,2
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LINKS
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FORMULA
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a(n) = Sum_{k=0..n} binomial(2*k,k)*binomial(2*n,n-k).
First column of A094527^2. 1 + x*exp( Sum_{n >= 1} a(n)*x^n/n ) = 1 + x + 4*x^2 + 18*x^3 + 86*x^4 + ... is the o.g.f. for A153294. - Peter Bala, Jul 21 2015
Conjecture D-finite with recurrence: 2*n*(2*n-1)*(3*n-5)*a(n) +(-123*n^3+328*n^2-249*n+60)*a(n-1) +50*(n-1)*(2*n-3)*(3*n-2)*a(n-2)=0. - R. J. Mathar, Jun 14 2016
a(n) = binomial(2*n, n)*hypergeom([1/2, -n], [n + 1], -4]. - Peter Luschny, Aug 04 2019
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MATHEMATICA
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CoefficientList[Series[1/(Sqrt[(1-4*x)*(2*Sqrt[1-4*x]+5*x-2)/x]), {x, 0, 20}], x] (* Vaclav Kotesovec, Jun 08 2014 *)
A243585[n_] := Binomial[2 n, n] Hypergeometric2F1[1/2, -n, n + 1, -4];
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PROG
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(Maxima)
a(n):=sum(binomial(2*k, k)*binomial(2*n, n-k), k, 0, n);
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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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