A004311 Binomial coefficient C(2n,n-5).

1, 12, 91, 560, 3060, 15504, 74613, 346104, 1562275, 6906900, 30045015, 129024480, 548354040, 2310789600, 9669554100, 40225345056, 166509721602, 686353797976, 2818953098830, 11541847896480, 47129212243960, 191991813933920, 780512175396135, 3167295784216200

Offset: 5

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=5. - 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 = 5..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 [math.CO], 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.
• Franck Ramaharo, Statistics on some classes of knot shadows, arXiv:1802.07701 [math.CO], 2018.

Crossrefs

Diagonal 11 of triangle A100257.

Cf. A001622.

Programs

Magma

[ Binomial(2*n,n-5): n in [5..150] ]; // Vincenzo Librandi, Apr 13 2011

Mathematica

Table[Binomial[2*n, n-5], {n, 5, 30}] (* Amiram Eldar, Aug 27 2022 *)

PARI

first(m)=vector(m,i,binomial(2*(i+4),i-1)) \\ Anders Hellström, Aug 17 2015

Formula

a(n) = Sum{k=0..n} C(n, k)*C(n, k+5). - Hermann Stamm-Wilbrandt, Aug 17 2015

-(n-5)*(n+5)*a(n) +2*n*(2*n-1)*a(n-1)=0. - R. J. Mathar, Jan 24 2018

E.g.f.: BesselI(5,2*x) * exp(2*x). - Ilya Gutkovskiy, Jun 27 2019

From Amiram Eldar, Aug 27 2022: (Start)

Sum_{n>=5} 1/a(n) = 6169/840 - 31*Pi/(9*sqrt(3)).

Sum_{n>=5} (-1)^(n+1)/a(n) = 5254*log(phi)/(5*sqrt(5)) - 63059/280, where phi is the golden ratio (A001622). (End)

G.f.: 2F1([11/2,6],[11],4*x). - Karol A. Penson, Apr 24 2024