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A051960
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a(n) = C(n)*(3n+2) where C(n) = Catalan numbers = A000108.
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9
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2, 5, 16, 55, 196, 714, 2640, 9867, 37180, 140998, 537472, 2057510, 7904456, 30458900, 117675360, 455657715, 1767883500, 6871173870, 26747767200, 104268528210, 406975466040, 1590307356300, 6220814327520, 24357232569150, 95452906901976, 374369872911804
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OFFSET
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0,1
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COMMENTS
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If Y is a fixed 2-subset of a 2n-set X then a(n-1) is the number of n-subsets of X intersecting Y. - Milan Janjic, Oct 21 2007
a(n-1) is the number of vertices in the n-dimensional halohedron (or equivalently, n-cycle cubeahedron). - Vincent Pilaud, May 12 2020
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REFERENCES
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Albert H. Beiler, Recreations in the Theory of Numbers, Dover, N.Y., 1964, pp. 194-196.
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LINKS
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FORMULA
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(n+1)*a(n) - 2*(n+2)*a(n-1) - 4*(2*n-3)*a(n-2) = 0. - conjectured by R. J. Mathar, Oct 02 2014, verified by Robert Israel, Sep 30 2015
a(n) = Sum_{k=0..(n+1)/2} (binomial(n-k+1,k)*2^(n-2*k+1)*binomial(n,k)). - Vladimir Kruchinin, Sep 30 2015.
a(n) = 4^n*(2+3*n)*Gamma(n + 1/2)/(sqrt(Pi)*Gamma(n+2)). - Peter Luschny, Dec 14 2015
Sum_{n>=0} a(n)/8^n = 5*sqrt(2) - 4. - Amiram Eldar, May 06 2023
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MAPLE
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a := n -> 4^n*(2+3*n)*GAMMA(1/2+n)/(sqrt(Pi)*GAMMA(2+n)):
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MATHEMATICA
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PROG
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(Maxima)
a(n):=sum(binomial(n-k+1, k)*2^(n-2*k+1)*binomial(n, k), k, 0, (n+1)/2); /* Vladimir Kruchinin, Sep 30 2015 */
(PARI) a(n) = (3*n+2)*binomial(2*n, n)/(n+1);
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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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