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%I #59 Oct 24 2023 21:52:07
%S 1,10,66,364,1820,8568,38760,170544,735471,3124550,13123110,54627300,
%T 225792840,927983760,3796297200,15471286560,62852101650,254661927156,
%U 1029530696964,4154246671960,16735679449896,67327446062800,270533919634160,1085929983159840,4355031703297275
%N Binomial coefficient C(2n,n-4).
%C 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 Seiichi Manyama, <a href="/A004310/b004310.txt">Table of n, a(n) for n = 4..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], p 828
%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 [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.
%H 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 -(n-4)*(n+4)*a(n) +2*n*(2*n-1)*a(n-1)=0. - _R. J. Mathar_, Dec 22 2013
%F 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 a(n) = Sum_{k=0..n} C(n, k)*C(n, k+4). - _Hermann Stamm-Wilbrandt_, Aug 17 2015
%F E.g.f.: BesselI(4,2*x)*exp(2*x). - _Ilya Gutkovskiy_, Jun 27 2019
%F From _Amiram Eldar_, Aug 27 2022: (Start)
%F Sum_{n>=4} 1/a(n) = 23*Pi/(9*sqrt(3)) - 211/60.
%F Sum_{n>=4} (-1)^n/a(n) = 1586*log(phi)/(5*sqrt(5)) - 1347/20, where phi is the golden ratio (A001622). (End)
%t Table[Binomial[2*n, n-4], {n, 4, 30}] (* _Amiram Eldar_, Aug 27 2022 *)
%o (Magma) [ Binomial(2*n,n-4): n in [4..150] ]; // _Vincenzo Librandi_, Apr 13 2011
%o (PARI) first(m)=vector(m,i,binomial(2*(i+3),i-1)) \\ _Anders Hellström_, Aug 17 2015
%Y Diagonal 9 of triangle A100257.
%Y Cf. A001622.
%K nonn,easy
%O 4,2
%A _N. J. A. Sloane_
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