A004314 a(n) = binomial coefficient C(2n, n - 8).

1, 18, 190, 1540, 10626, 65780, 376740, 2035800, 10518300, 52451256, 254186856, 1203322288, 5586853480, 25518731280, 114955808528, 511738760544, 2254848913647, 9847379391150, 42671977361650, 183649923622620, 785613562163430, 3342649210440540, 14154280149473100

Offset: 8

Author

N. J. A. Sloane

Keywords

• nonn
• easy
• changed

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 = 8..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.

Crossrefs

Cf. A001622.

Programs

Magma

[Binomial(2*n, n - 8): n in [8..150] ]; // Vincenzo Librandi, Apr 13 2011

Mathematica

a[n_]:=Binomial[2*n, n - 8]; Array[a, 150, 8] (* Stefano Spezia, Sep 01 2018 *)

PARI

a(n)=binomial(2*n,n-8) \\ Charles R Greathouse IV, Oct 23 2023

Formula

-(n - 8)*(n + 8)*a(n) + 2*n*(2*n - 1)*a(n - 1) = 0. - R. J. Mathar, Dec 10 2013

E.g.f.: BesselI(8,2*x) * exp(2*x). - Ilya Gutkovskiy, Jun 27 2019

From Amiram Eldar, Aug 27 2022: (Start)

Sum_{n>=8} 1/a(n) = 3941153/360360 - 49*Pi/(9*sqrt(3)).

Sum_{n>=8} (-1)^n/a(n) = 153506*log(phi)/(5*sqrt(5)) - 2380569277/360360, where phi is the golden ratio (A001622). (End)