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%I #50 Sep 03 2023 08:42:53
%S 1,12,91,560,3060,15504,74613,346104,1562275,6906900,30045015,
%T 129024480,548354040,2310789600,9669554100,40225345056,166509721602,
%U 686353797976,2818953098830,11541847896480,47129212243960,191991813933920,780512175396135,3167295784216200
%N Binomial coefficient C(2n,n-5).
%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=5. - _Herbert Kociemba_, May 24 2004
%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="/A004311/b004311.txt">Table of n, a(n) for n = 5..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="http://www.pmfbl.org/janjic/">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 a(n) = Sum{k=0..n} C(n, k)*C(n, k+5). - _Hermann Stamm-Wilbrandt_, Aug 17 2015
%F -(n-5)*(n+5)*a(n) +2*n*(2*n-1)*a(n-1)=0. - _R. J. Mathar_, Jan 24 2018
%F E.g.f.: BesselI(5,2*x)*exp(2*x). - _Ilya Gutkovskiy_, Jun 27 2019
%F From _Amiram Eldar_, Aug 27 2022: (Start)
%F Sum_{n>=5} 1/a(n) = 6169/840 - 31*Pi/(9*sqrt(3)).
%F Sum_{n>=5} (-1)^(n+1)/a(n) = 5254*log(phi)/(5*sqrt(5)) - 63059/280, where phi is the golden ratio (A001622). (End)
%t Table[Binomial[2*n, n-5], {n, 5, 30}] (* _Amiram Eldar_, Aug 27 2022 *)
%o (Magma) [ Binomial(2*n,n-5): n in [5..150] ]; // _Vincenzo Librandi_, Apr 13 2011
%o (PARI) first(m)=vector(m,i,binomial(2*(i+4),i-1)) \\ _Anders Hellström_, Aug 17 2015
%Y Diagonal 11 of triangle A100257.
%Y Cf. A001622.
%K nonn,easy
%O 5,2
%A _N. J. A. Sloane_
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