A004310 Binomial coefficient C(2n,n-4).

1, 10, 66, 364, 1820, 8568, 38760, 170544, 735471, 3124550, 13123110, 54627300, 225792840, 927983760, 3796297200, 15471286560, 62852101650, 254661927156, 1029530696964, 4154246671960, 16735679449896, 67327446062800, 270533919634160, 1085929983159840, 4355031703297275

Offset: 4

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=4. - Herbert Kociemba, May 23 2004

Links

• Seiichi Manyama, Table of n, a(n) for n = 4..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], p 828
• 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 9 of triangle A100257.

Cf. A001622.

Programs

Magma

[ Binomial(2*n,n-4): n in [4..150] ]; // Vincenzo Librandi, Apr 13 2011

Mathematica

Table[Binomial[2*n, n-4], {n, 4, 30}] (* Amiram Eldar, Aug 27 2022 *)

PARI

first(m)=vector(m,i,binomial(2*(i+3),i-1)) \\ Anders Hellström, Aug 17 2015

Formula

D-finite with recurrence -(n-4)*(n+4)*a(n) +2*n*(2*n-1)*a(n-1)=0. - R. J. Mathar, Dec 22 2013

G.f.: x*(1/(sqrt(1-4*x)*x)-(1-sqrt(1-4*x))/(2*x^2))/((1-sqrt(1-4*x))/(2*x)-1)^5-(1/x^4-6/x^3+10/x^2-4/x). - Vladimir Kruchinin, Aug 11 2015

a(n) = Sum_{k=0..n} C(n, k)*C(n, k+4). - Hermann Stamm-Wilbrandt, Aug 17 2015

E.g.f.: BesselI(4,2*x) * exp(2*x). - Ilya Gutkovskiy, Jun 27 2019

From Amiram Eldar, Aug 27 2022: (Start)

Sum_{n>=4} 1/a(n) = 23*Pi/(9*sqrt(3)) - 211/60.

Sum_{n>=4} (-1)^n/a(n) = 1586*log(phi)/(5*sqrt(5)) - 1347/20, where phi is the golden ratio (A001622). (End)

G.f.: 2F1([9/2,5],[9],4*x). - Karol A. Penson, Apr 24 2024