This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A004310 #71 Aug 24 2025 04:17:12
%S A004310 1,10,66,364,1820,8568,38760,170544,735471,3124550,13123110,54627300,
%T A004310 225792840,927983760,3796297200,15471286560,62852101650,254661927156,
%U A004310 1029530696964,4154246671960,16735679449896,67327446062800,270533919634160,1085929983159840,4355031703297275
%N A004310 Binomial coefficient C(2n,n-4).
%C A004310 Number of lattice paths from (0,0) to (n,n) with steps E=(1,0) and N=(0,1) which touch or cross the line x-y=4. - _Herbert Kociemba_, May 23 2004
%H A004310 Seiichi Manyama, <a href="/A004310/b004310.txt">Table of n, a(n) for n = 4..1000</a>
%H A004310 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], p 828
%H A004310 Milan Janjic, <a href="https://pmf.unibl.org/wp-content/uploads/2017/10/enumfor.pdf">Two Enumerative Functions</a>
%H A004310 Milan Janjic and B. Petkovic, <a href="http://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 A004310 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.
%H A004310 Franck Ramaharo, <a href="https://arxiv.org/abs/1802.07701">Statistics on some classes of knot shadows</a>, arXiv:1802.07701 [math.CO], 2018.
%F A004310 D-finite with recurrence -(n-4)*(n+4)*a(n) +2*n*(2*n-1)*a(n-1)=0. - _R. J. Mathar_, Dec 22 2013
%F A004310 G.f.: x*(1/(sqrt(1-4*x)*x)-(1-sqrt(1-4*x))/(2*x^2))/((1-sqrt(1-4*x))/(2*x)-1)^5-(1/x^4-6/x^3+10/x^2-4/x). - _Vladimir Kruchinin_, Aug 11 2015
%F A004310 a(n) = Sum_{k=0..n} C(n, k)*C(n, k+4). - _Hermann Stamm-Wilbrandt_, Aug 17 2015
%F A004310 E.g.f.: BesselI(4,2*x) * exp(2*x). - _Ilya Gutkovskiy_, Jun 27 2019
%F A004310 From _Amiram Eldar_, Aug 27 2022: (Start)
%F A004310 Sum_{n>=4} 1/a(n) = 23*Pi/(9*sqrt(3)) - 211/60.
%F A004310 Sum_{n>=4} (-1)^n/a(n) = 1586*log(phi)/(5*sqrt(5)) - 1347/20, where phi is the golden ratio (A001622). (End)
%F A004310 G.f.: 2F1([9/2,5],[9],4*x). - _Karol A. Penson_, Apr 24 2024
%t A004310 Table[Binomial[2*n, n-4], {n, 4, 30}] (* _Amiram Eldar_, Aug 27 2022 *)
%o A004310 (Magma) [ Binomial(2*n,n-4): n in [4..150] ]; // _Vincenzo Librandi_, Apr 13 2011
%o A004310 (PARI) first(m)=vector(m,i,binomial(2*(i+3),i-1)) \\ _Anders Hellström_, Aug 17 2015
%Y A004310 Diagonal 9 of triangle A100257.
%Y A004310 Cf. A001622.
%K A004310 nonn,easy,changed
%O A004310 4,2
%A A004310 _N. J. A. Sloane_