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
Seiichi Manyama, Table of n, a(n) for n = 6..1000
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, Two Enumerative Functions
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
KEYWORD
nonn,easy
AUTHOR
STATUS
approved