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A004312
Binomial coefficient C(2n,n-6).
4
1, 14, 120, 816, 4845, 26334, 134596, 657800, 3108105, 14307150, 64512240, 286097760, 1251677700, 5414950296, 23206929840, 98672427616, 416714805914, 1749695026860, 7309837001104, 30405943383200, 125994627894135, 520341450264090, 2142582442263900, 8799226775309880
OFFSET
6,2
COMMENTS
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
REFERENCES
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.
LINKS
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].
Milan Janjic and B. Petkovic, A Counting Function, arXiv preprint arXiv:1301.4550, 2013. - From N. J. A. Sloane, Feb 13 2013
Milan Janjic and B. Petkovic, A Counting Function Generalizing Binomial Coefficients and Some Other Classes of Integers, J. Int. Seq. 17 (2014), Article 14.3.5.
FORMULA
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
E.g.f.: BesselI(6,2*x)*exp(2*x). - Ilya Gutkovskiy, Jun 27 2019
From Amiram Eldar, Aug 27 2022: (Start)
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)
MATHEMATICA
Table[Binomial[2*n, n-6], {n, 6, 30}] (* Amiram Eldar, Aug 27 2022 *)
PROG
(Magma) [ Binomial(2*n, n-6): n in [6..150] ]; // Vincenzo Librandi, Apr 13 2011
(PARI) a(n)=binomial(2*n, n-6) \\ Charles R Greathouse IV, Oct 23 2023
CROSSREFS
Diagonal 13 of triangle A100257.
Cf. A001622.
Sequence in context: A240051 A202072 A280003 * A002056 A249980 A206635
KEYWORD
nonn,easy
STATUS
approved