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A028329
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Twice central binomial coefficients.
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23
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2, 4, 12, 40, 140, 504, 1848, 6864, 25740, 97240, 369512, 1410864, 5408312, 20801200, 80233200, 310235040, 1202160780, 4667212440, 18150270600, 70690527600, 275693057640, 1076515748880, 4208197927440, 16466861455200, 64495207366200, 252821212875504, 991837065896208
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OFFSET
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0,1
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COMMENTS
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Central elements in the even-Pascal triangle A028326.
If Y is a 3-subset of an 2n-set X then, for n>=3, a(n-1) is the number of (n+1)-subsets of X having at least two elements in common with Y. - Milan Janjic, Dec 16 2007
a(n) denotes the number of ways one can reach the (n,n) point in an n X n grid via the point (n-1, n-1) starting from (0,0) when moving right and up is allowed [From Avik Roy (avik_3.1416(AT)yahoo.co.in), Jan 29 2009]
It appears that a(n-1) is also the number of quivers in the mutation class of twisted types BD_n and CD_n for n >= 3. - Christian Stump, Nov 03 2010
This is the case m = n+1 in the Catalan's formula (2m)!*(2n)!/(m!*(m+n)!*n!) - see Umberto Scarpis in References. - Bruno Berselli, Apr 27 2012
a(n) is the number of North-East paths from (0,0) to (n+1,n+1) that bounce off the diagonal y = x an even number of times. Details can be found in Section 4.2 in Pan and Remmel's link.
a(n) is the number of North-East paths from (0,0) to (n+1,n+1) that cross the diagonal y = x an even number of times. Details can be found in Section 4.3 in Pan and Remmel's link. (End)
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REFERENCES
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Umberto Scarpis, Sui numeri primi e sui problemi dell'analisi indeterminata in Questioni riguardanti le matematiche elementari, Nicola Zanichelli Editore (1924-1927, third Edition), page 11.
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LINKS
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FORMULA
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G.f.: 2/sqrt(1 - 4*x).
a(n) = 2 * binomial(2*n, n).
G.f.: G(0), where G(k)= 1 + 1/(1 - 2*x*(2*k + 1)/(2*x*(2*k + 1) + (k + 1)/ G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Jun 07 2013
D-finite with recurrence: n*a(n) + 2*(-2*n+1)*a(n-1)=0. - R. J. Mathar, Jan 17 2020
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MAPLE
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seq(add(binomial(2*n, n), k=1..2), n=0..23); # Zerinvary Lajos, Dec 14 2007
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MATHEMATICA
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PROG
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(PARI) a(n)=2*binomial(2*n, n)
(Magma) [2*(n+1)*Catalan(n): n in [0..30]]; // G. C. Greubel, Jul 13 2024
(SageMath) [2*binomial(2*n, n) for n in range(31)] # G. C. Greubel, Jul 13 2024
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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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