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 A004312 #51 Aug 24 2025 04:18:45
%S A004312 1,14,120,816,4845,26334,134596,657800,3108105,14307150,64512240,
%T A004312 286097760,1251677700,5414950296,23206929840,98672427616,416714805914,
%U A004312 1749695026860,7309837001104,30405943383200,125994627894135,520341450264090,2142582442263900,8799226775309880
%N A004312 Binomial coefficient C(2n,n-6).
%C A004312 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=6. - _Herbert Kociemba_, May 24 2004
%D A004312 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 A004312 Seiichi Manyama, <a href="/A004312/b004312.txt">Table of n, a(n) for n = 6..1000</a>
%H A004312 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 A004312 Milan Janjic, <a href="https://pmf.unibl.org/wp-content/uploads/2017/10/enumfor.pdf">Two Enumerative Functions</a>
%H A004312 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 A004312 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 A004312 G.f.: ((1/(sqrt(1-4*x)*x)-(1-sqrt(1-4*x))/(2*x^2))*x)/((1-sqrt(1-4*x))/(2*x)-1)^7+6/x-35/x^2+56/x^3-36/x^4+10/x^5-1/x^6. - _Vladimir Kruchinin_, Aug 11 2015
%F A004312 -(n-6)*(n+6)*a(n) +2*n*(2*n-1)*a(n-1)=0. - _R. J. Mathar_, Jan 24 2018
%F A004312 E.g.f.: BesselI(6,2*x) * exp(2*x). - _Ilya Gutkovskiy_, Jun 27 2019
%F A004312 From _Amiram Eldar_, Aug 27 2022: (Start)
%F A004312 Sum_{n>=6} 1/a(n) = 2*Pi/(9*sqrt(3)) + 1709/2520.
%F A004312 Sum_{n>=6} (-1)^n/a(n) = 16636*log(phi)/(5*sqrt(5)) - 1802033/2520, where phi is the golden ratio (A001622). (End)
%t A004312 Table[Binomial[2*n, n-6], {n, 6, 30}] (* _Amiram Eldar_, Aug 27 2022 *)
%o A004312 (Magma) [ Binomial(2*n,n-6): n in [6..150] ]; // _Vincenzo Librandi_, Apr 13 2011
%o A004312 (PARI) a(n)=binomial(2*n,n-6) \\ _Charles R Greathouse IV_, Oct 23 2023
%Y A004312 Diagonal 13 of triangle A100257.
%Y A004312 Cf. A001622.
%K A004312 nonn,easy,changed
%O A004312 6,2
%A A004312 _N. J. A. Sloane_