A004315 a(n) = binomial coefficient C(2n, n-9).

1, 20, 231, 2024, 14950, 98280, 593775, 3365856, 18156204, 94143280, 472733756, 2311801440, 11058116888, 51915526432, 239877544005, 1093260079344, 4923689695575, 21945588357420, 96926348578605, 424655979547800, 1847253511032930, 7984465725343800, 34315056105966195

Offset: 9

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 = 9..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-9): n in [9..150] ]; // Vincenzo Librandi, Apr 13 2011

Mathematica

Table[Binomial[2*n, n-9], {n, 9, 30}] (* Amiram Eldar, Aug 27 2022 *)

PARI

a(n)=binomial(2*n,n-9) \\ Charles R Greathouse IV, Oct 23 2023

Formula

-(n-9)*(n+9)*a(n) + 2*n*(2*n-1)*a(n-1) = 0. - R. J. Mathar, Dec 10 2013

E.g.f.: BesselI(9,2*x) * exp(2*x). - Ilya Gutkovskiy, Jun 27 2019

From Amiram Eldar, Aug 27 2022: (Start)

Sum_{n>=9} 1/a(n) = 2*Pi/(9*sqrt(3)) + 7457/11440.

Sum_{n>=9} (-1)^(n+1)/a(n) = 453564*log(phi)/(5*sqrt(5)) - 14069064271/720720, where phi is the golden ratio (A001622). (End)