A004316 a(n) = binomial coefficient C(2n, n-10).

1, 22, 276, 2600, 20475, 142506, 906192, 5379616, 30260340, 163011640, 847660528, 4280561376, 21090682613, 101766230790, 482320623240, 2250829575120, 10363194502115, 47153358767970, 212327989773900, 947309492837400, 4191844505805495, 18412956934908690, 80347448443237920

Offset: 10

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 = 10..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-10): n in [10..150] ]; // Vincenzo Librandi, Apr 13 2011

Mathematica

Table[Binomial[2*n, n-10], {n, 10, 30}] (* Amiram Eldar, Aug 27 2022 *)

PARI

a(n)=binomial(2*n,n-10) \\ Charles R Greathouse IV, Oct 23 2023

Formula

E.g.f.: BesselI(10,2*x) * exp(2*x). - Ilya Gutkovskiy, Jun 27 2019

From Amiram Eldar, Aug 27 2022: (Start)

Sum_{n>=10} 1/a(n) = 59*Pi/(9*sqrt(3)) - 26565167/2450448.

Sum_{n>=10} (-1)^n/a(n) = 1322746*log(phi)/(5*sqrt(5)) - 697534881193/12252240, where phi is the golden ratio (A001622). (End)

D-finite with recurrence -(n-10)*(n+10)*a(n) +2*n*(2*n-1)*a(n-1)=0. - R. J. Mathar, Jan 13 2025