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A276666
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a(n) = (n-1)*Catalan(n).
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2
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-1, 0, 2, 10, 42, 168, 660, 2574, 10010, 38896, 151164, 587860, 2288132, 8914800, 34767720, 135727830, 530365050, 2074316640, 8119857900, 31810737420, 124718287980, 489325340400, 1921133836440, 7547311500300, 29667795388452, 116686713634848, 459183826803800
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OFFSET
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0,3
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LINKS
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FORMULA
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a(n) = [x^n] (1-3*x)/(x*sqrt(1-4*x))-1/x.
a(n) = 4^n*(n-1)*hypergeom([3/2, -n], [2], 1).
a(n) = 4^n*(n-1)*JacobiP(n,1,-1/2-n,-1)/(n+1).
a(n) = (2*n)! [x^(2^n)]( BesselI(2,2*x) - (1+1/x)*BesselI(1,2*x) ).
a(n) = A056040(2*n)*(1-2/(n+1)) = (n^2-1)*(2*n)!/(n+1)!^2.
a(n) = a(n-1)*2*(n-1)*(2*n-1)/((n-2)*(n+1)) for n > 2. - Chai Wah Wu, Sep 12 2016
Sum_{n>=2} 1/a(n) = 5/6 - Pi/(9*sqrt(3)).
Sum_{n>=2} (-1)^n/a(n) = 26*sqrt(5)*log(phi)/25 - 7/10, where phi is the golden ratio (A001622). (End)
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MAPLE
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f := (1-3*x)/(x*sqrt(1-4*x))-1/x:
series(f, x, 29): seq(coeff(%, x, n), n=0..26);
A276666 := n -> (n^2-1)*(2*n)!/(n+1)!^2:
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MATHEMATICA
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PROG
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(Sage)
A276666 = lambda n: (n - 1) * catalan_number(n)
(PARI) a(n) = if(n==0, -1, 2*binomial(2*n-1, n+1)); \\ G. C. Greubel, Aug 29 2019
(GAP) Concatenation([-1], List([1..30], n-> 2*Binomial(2*n-1, n+1))); # G. C. Greubel, Aug 29 2019
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CROSSREFS
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A024483 is a variant of this sequence.
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KEYWORD
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sign
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AUTHOR
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STATUS
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approved
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