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A004310
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Binomial coefficient C(2n,n-4).
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4
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1, 10, 66, 364, 1820, 8568, 38760, 170544, 735471, 3124550, 13123110, 54627300, 225792840, 927983760, 3796297200, 15471286560, 62852101650, 254661927156, 1029530696964, 4154246671960, 16735679449896, 67327446062800, 270533919634160, 1085929983159840, 4355031703297275
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OFFSET
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4,2
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COMMENTS
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Number of lattice paths from (0,0) to (n,n) with steps E=(1,0) and N=(0,1) which touch or cross the line x-y=4. - Herbert Kociemba, May 23 2004
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LINKS
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M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy], p 828
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FORMULA
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-(n-4)*(n+4)*a(n) +2*n*(2*n-1)*a(n-1)=0. - R. J. Mathar, Dec 22 2013
G.f.: x*(1/(sqrt(1-4*x)*x)-(1-sqrt(1-4*x))/(2*x^2))/((1-sqrt(1-4*x))/(2*x)-1)^5-(1/x^4-6/x^3+10/x^2-4/x). - Vladimir Kruchinin, Aug 11 2015
Sum_{n>=4} 1/a(n) = 23*Pi/(9*sqrt(3)) - 211/60.
Sum_{n>=4} (-1)^n/a(n) = 1586*log(phi)/(5*sqrt(5)) - 1347/20, where phi is the golden ratio (A001622). (End)
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MATHEMATICA
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Table[Binomial[2*n, n-4], {n, 4, 30}] (* Amiram Eldar, Aug 27 2022 *)
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PROG
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(PARI) first(m)=vector(m, i, binomial(2*(i+3), i-1)) \\ Anders Hellström, Aug 17 2015
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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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STATUS
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approved
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