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A004312
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Binomial coefficient C(2n,n-6).
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4
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1, 14, 120, 816, 4845, 26334, 134596, 657800, 3108105, 14307150, 64512240, 286097760, 1251677700, 5414950296, 23206929840, 98672427616, 416714805914, 1749695026860, 7309837001104, 30405943383200, 125994627894135, 520341450264090, 2142582442263900, 8799226775309880
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OFFSET
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6,2
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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=6. - Herbert Kociemba, May 24 2004
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REFERENCES
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M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 828.
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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].
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FORMULA
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G.f.: ((1/(sqrt(1-4*x)*x)-(1-sqrt(1-4*x))/(2*x^2))*x)/((1-sqrt(1-4*x))/(2*x)-1)^7+6/x-35/x^2+56/x^3-36/x^4+10/x^5-1/x^6. - Vladimir Kruchinin, Aug 11 2015
-(n-6)*(n+6)*a(n) +2*n*(2*n-1)*a(n-1)=0. - R. J. Mathar, Jan 24 2018
Sum_{n>=6} 1/a(n) = 2*Pi/(9*sqrt(3)) + 1709/2520.
Sum_{n>=6} (-1)^n/a(n) = 16636*log(phi)/(5*sqrt(5)) - 1802033/2520, where phi is the golden ratio (A001622). (End)
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MATHEMATICA
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Table[Binomial[2*n, n-6], {n, 6, 30}] (* Amiram Eldar, Aug 27 2022 *)
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PROG
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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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