A004312 Binomial coefficient C(2n,n-6).

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

Author

N. J. A. Sloane

Keywords

• nonn
• easy
• changed

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.

Crossrefs

Diagonal 13 of triangle A100257.

Cf. A001622.

Programs

Magma

[ Binomial(2*n,n-6): n in [6..150] ]; // Vincenzo Librandi, Apr 13 2011

Mathematica

Table[Binomial[2*n, n-6], {n, 6, 30}] (* Amiram Eldar, Aug 27 2022 *)

PARI

a(n)=binomial(2*n,n-6) \\ Charles R Greathouse IV, Oct 23 2023

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)