A004313 - OEIS

%I #49 Nov 05 2025 15:21:44

%S 1,16,153,1140,7315,42504,230230,1184040,5852925,28048800,131128140,

%T 600805296,2707475148,12033222880,52860229080,229911617056,

%U 991493848554,4244421484512,18053528883775,76360380541900,321387366339585,1346766106565880,5621728217559090

%N a(n) = binomial coefficient C(2n, n-7).

%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="/A004313/b004313.txt">Table of n, a(n) for n = 7..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="https://arxiv.org/abs/1301.4550">A Counting Function</a>, arXiv preprint arXiv:1301.4550 [math.CO], 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 -(n-7)*(n+7)*a(n) + 2*n*(2*n-1)*a(n-1) = 0. - _R. J. Mathar_, Jan 24 2018

%F E.g.f.: BesselI(7,2*x)*exp(2*x). - _Ilya Gutkovskiy_, Jun 27 2019

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

%F Sum_{n>=7} 1/a(n) = 41*Pi/(9*sqrt(3)) - 24923/3465.

%F Sum_{n>=7} (-1)^(n+1)/a(n) = 51094*log(phi)/(5*sqrt(5)) - 7616722/3465, where phi is the golden ratio (A001622). (End)

%t Table[Binomial[2n,n-7],{n,7,30}] (* _Harvey P. Dale_, Nov 27 2013 *)

%o (Magma) [ Binomial(2*n,n-7): n in [7..150] ]; // _Vincenzo Librandi_, Apr 13 2011

%o (PARI) a(n)=binomial(2*n,n-7) \\ _Charles R Greathouse IV_, Oct 23 2023

%Y Diagonal 15 of triangle A100257.

%Y Cf. A001622.

%K nonn,easy

%O 7,2

%A _N. J. A. Sloane_