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A037965
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a(n) = n*binomial(2*n-2, n-1).
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14
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0, 1, 4, 18, 80, 350, 1512, 6468, 27456, 115830, 486200, 2032316, 8465184, 35154028, 145608400, 601749000, 2481880320, 10218366630, 42004911960, 172427570700, 706905276000, 2894777105220, 11841673237680, 48394276165560, 197602337462400, 806190092077500
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OFFSET
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0,3
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COMMENTS
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REFERENCES
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The right-hand side of a binomial coefficient identity in H. W. Gould, Combinatorial Identities, Morgantown, 1972.
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LINKS
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FORMULA
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G.f.: Hypergeometric2F1([1/2, 2], [1], 4*x). - Paul Barry, Sep 03 2008
G.f.: x*(1-2*x)/(1-4*x)^(3/2);
a(n+1) = Sum_{k=0..n} binomial(2*k,k)*(4^(n-k) + 0^(n-k))/2. (End)
D-finite with recurrence (n-1)*a(n) - 2*(3*n-4)*a(n-1) + 4*(2*n-5)*a(n-2) = 0. - R. J. Mathar, Nov 30 2012
Sum_{n>=1} 1/a(n) = 4*Pi/(3*sqrt(3)) - Pi^2/9.
Sum_{n>=1} (-1)^(n+1)/a(n) = 8*log(phi)/sqrt(5) - 4*log(phi)^2, where phi is the golden ratio (A001622). (End)
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MATHEMATICA
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a[n_]:= n*Binomial[2*n-2, n-1]; Array[a, 30, 0] (* Amiram Eldar, Mar 10 2022 *)
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PROG
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(PARI) a(n) = n*binomial(2*n-2, n-1); \\ Joerg Arndt, Sep 04 2017
(Magma) [0] cat [n^2*Catalan(n-1): n in [1..30]]; // G. C. Greubel, Jun 19 2022
(SageMath) [n^2*catalan_number(n-1) for n in (0..30)] # G. C. Greubel, Jun 19 2022
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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