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A249520
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Expansion of: G.f.: 2-(2*x)/(1-sqrt(sqrt(1-16*x)+1)/sqrt(2)).
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1
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1, 5, 17, 134, 1333, 14890, 178394, 2240268, 29101197, 387793090, 5271628846, 72818272852, 1019157450818, 14421479205284, 205978607191508, 2965567162041368, 42994130150806077, 627124332791860146, 9196734644381065510, 135516564162069215748, 2005456310676742385910
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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_{i=0..n} 2^i*binomial(2*(n-1)+i-1,i)*binomial(2*n-i-2,n-i))/(n-1),n>1, a(0)=1, a(1)=5.
a(n) ~ (3+2*sqrt(2)) * 2^(4*n-9/2) / (sqrt(Pi) * n^(3/2)). - Vaclav Kotesovec, Oct 31 2014
D-finite with recurrence: n*(n-1)*(2*n-3)*a(n) -2*(n-1)*(32*n^2-128*n+135)*a(n-1) +16*(2*n-5)*(4*n-9)*(4*n-11)*a(n-2)=0. - R. J. Mathar, Jan 25 2020
a(n) = 2^(2*n-1)*C(n-1) + C(2*n-1) + 2*C(2*n-2), for n>0, where C(n) is the n-th Catalan number, A000108. - Ira M. Gessel, Dec 10 2020
a(n) = (Binomial(2*n - 2, n)*hypergeom([-n, 2*n - 2], [2 - 2*n], 2])/(n - 1) for n >= 2. - Peter Luschny, Dec 10 2020
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MAPLE
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a := n -> `if`(n=0, 1, `if`(n=1, 5, ((4^(2*n-1))/((+2*n-1)*(4*n-3)*(4*n-1) *Pi*GAMMA(1+2*n)))*((6*sqrt(Pi)*(1-2*n)^2*GAMMA(2*n+1/2)+4^n*(4*n-3)*(4*n-1)*GAMMA(n+1/2)^2)))): seq(a(n), n=0..18); # Peter Luschny, Oct 31 2014
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MATHEMATICA
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CoefficientList[Series[2-(2*x)/(1-Sqrt[Sqrt[1-16*x]+1]/Sqrt[2]), {x, 0, 20}], x] (* Vaclav Kotesovec, Oct 31 2014 *)
a[n_] := (Binomial[2n - 2, n] Hypergeometric2F1[-n, 2n - 2, 2 - 2n, 2])/(n - 1);
a[0] := 1; a[1] := 5; Table[a[n], {n, 0, 18}] (* Peter Luschny, Dec 10 2020 *)
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PROG
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(Maxima)
a(n)=if n=0 then 1 else if n=1 then 5 else sum(2^i*binomial(2*(n-1)+i-1, i)*binomial(2*n-i-2, n-i), i, 0, n)/(n-1);
(PARI) x='x+O('x^50); Vec(2 - (2*x)/(1-sqrt(sqrt(1-16*x)+1)/sqrt(2))) \\ G. C. Greubel, Jun 02 2017
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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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