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

5,2

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=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.

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)

Mathematica

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

Prog

(Magma) [ Binomial(2*n, n-5): n in [5..150] ]; // Vincenzo Librandi, Apr 13 2011
(PARI) first(m)=vector(m, i, binomial(2*(i+4), i-1)) \\ Anders Hellström, Aug 17 2015

Crossrefs

Diagonal 11 of triangle A100257.

Cf. A001622.

Sequence in context: A114860 A001502 A001503 * A160869 A026074 A339715

Adjacent sequences: A004308 A004309 A004310 * A004312 A004313 A004314

Keyword

nonn,easy

Author

N. J. A. Sloane

Status

approved