A004316 - OEIS

%I #38 Jan 13 2025 05:00:20

%S 1,22,276,2600,20475,142506,906192,5379616,30260340,163011640,

%T 847660528,4280561376,21090682613,101766230790,482320623240,

%U 2250829575120,10363194502115,47153358767970,212327989773900,947309492837400,4191844505805495,18412956934908690,80347448443237920

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

%D 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.

%H Seiichi Manyama, <a href="/A004316/b004316.txt">Table of n, a(n) for n = 10..1000</a>

%H M. Abramowitz and I. A. Stegun, eds., <a href="http://www.convertit.com/Go/ConvertIt/Reference/AMS55.ASP">Handbook of Mathematical Functions</a>, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].

%H Milan Janjic, <a href="https://pmf.unibl.org/wp-content/uploads/2017/10/enumfor.pdf">Two Enumerative Functions</a>

%H Milan Janjic and B. Petkovic, <a href="http://arxiv.org/abs/1301.4550">A Counting Function</a>, arXiv preprint arXiv:1301.4550, 2013. - From _N. J. A. Sloane_, Feb 13 2013

%H Milan Janjic and B. Petkovic, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL17/Janjic/janjic45.html">A Counting Function Generalizing Binomial Coefficients and Some Other Classes of Integers</a>, J. Int. Seq. 17 (2014), Article 14.3.5.

%F E.g.f.: BesselI(10,2*x)*exp(2*x). - _Ilya Gutkovskiy_, Jun 27 2019

%F From _Amiram Eldar_, Aug 27 2022: (Start)

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

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

%F 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

%t Table[Binomial[2*n, n-10], {n, 10, 30}] (* _Amiram Eldar_, Aug 27 2022 *)

%o (Magma) [ Binomial(2*n,n-10): n in [10..150] ]; // _Vincenzo Librandi_, Apr 13 2011

%o (PARI) a(n)=binomial(2*n,n-10) \\ _Charles R Greathouse IV_, Oct 23 2023

%Y Cf. A001622.

%K nonn,easy

%O 10,2

%A _N. J. A. Sloane_