%I M4532 N1921 #85 Oct 14 2024 10:56:36
%S 1,8,45,220,1001,4368,18564,77520,319770,1307504,5311735,21474180,
%T 86493225,347373600,1391975640,5567902560,22239974430,88732378800,
%U 353697121050,1408831480056,5608233007146,22314239266528,88749815264600,352870329957600,1402659561581460
%N Binomial coefficients C(2n,n-3).
%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=3. - _Herbert Kociemba_, May 23 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.
%D C. Lanczos, Applied Analysis. Prentice-Hall, Englewood Cliffs, NJ, 1956, p. 517.
%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H Robert Israel, <a href="/A002696/b002696.txt">Table of n, a(n) for n = 3..1497</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 A. Claesson and T. Mansour, <a href="http://arxiv.org/abs/math/0110036">Counting patterns of type (1,2) or (2,1)</a>, arXiv:math/0110036 [math.CO], 2001.
%H Milan Janjic, <a href="https://web.archive.org/web/20181001110015/https://pmf.unibl.org/wp-content/uploads/2017/10/enumfor.pdf">Two Enumerative Functions</a>
%H C. Lanczos, <a href="/A002457/a002457.pdf">Applied Analysis</a> (Annotated scans of selected pages)
%H Toufik Mansour and Mark Shattuck, <a href="https://doi.org/10.26493/2590-9770.1552.b43">Counting occurrences of subword patterns in non-crossing partitions</a>, Art Disc. Appl. Math. (2022).
%H R. Parviainen, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL9/Parviainen/parviainen3.html">Lattice Path Enumeration of Permutations with k Occurrences of the Pattern 2-13</a>, Journal of Integer Sequences, Vol. 9 (2006), Article 06.3.2.
%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.
%H Hermann Stamm-Wilbrandt, <a href="/A002696/a002696.gif">Compute C(2n, n-k) based on C(n,...) animation</a>
%H Daniel W. Stasiuk, <a href="http://hdl.handle.net/10388/11865">An Enumeration Problem for Sequences of n-ary Trees Arising from Algebraic Operads</a>, Master's Thesis, University of Saskatchewan-Saskatoon (2018).
%F G.f.: (1-sqrt(1-4*z))^6/(64*z^3*sqrt(1-4*z)). - _Emeric Deutsch_, Jan 28 2004
%F a(n) = Sum_{k=0..n} C(n, k)*C(n, k+3). - _Hermann Stamm-Wilbrandt_, Aug 17 2015
%F From _Robert Israel_, Aug 19 2015: (Start)
%F (n-2)*(n+4)*a(n+1) = (2*n+2)*(2*n+1)*a(n).
%F E.g.f.: I_3(2*x)*exp(2*x) where I_3 is a modified Bessel function. (End)
%F From _Amiram Eldar_, Aug 27 2022: (Start)
%F Sum_{n>=3} 1/a(n) = 3/4 + 2*Pi/(9*sqrt(3)).
%F Sum_{n>=3} (-1)^(n+1)/a(n) = 444*log(phi)/(5*sqrt(5)) - 1093/60, where phi is the golden ratio (A001622). (End)
%F G.f.: 2F1([7/2,4],[7],4*x). - _Karol A. Penson_, Apr 24 2024
%F From _Peter Bala_, Oct 13 2024: (Start)
%F a(n) = Integral_{x = 0..4} x^n * w(x) dx, where the weight function w(x) = 1/(2*Pi) * (x^3 - 6*x^2 + 9*x - 2)/sqrt(x*(4 - x)).
%F G.f: x^3 * B(x) * C(x)^6, where B(x) = 1/sqrt(1 - 4*x) is the g.f. of the central binomial coefficients A000984 and C(x) = (1 - sqrt(1 - 4*x))/(2*x) is the g.f. of the Catalan numbers A000108. (End)
%p A002696:=n->binomial(2*n,n-3): seq(A002696(n), n=3..30); # _Wesley Ivan Hurt_, Aug 19 2015
%t CoefficientList[Series[64/(((Sqrt[1-4x] +1)^6)*Sqrt[1-4x]), {x,0,30}], x] (* _Robert G. Wilson v_, Aug 08 2011 *)
%o (Magma) [ Binomial(2*n,n-3): n in [3..30] ]; // _Vincenzo Librandi_, Apr 13 2011
%o (PARI) a(n)=binomial(n+n,n-3) \\ _Charles R Greathouse IV_, Aug 08 2011
%o (Sage) [binomial(2*n, n-3) for n in (3..30)] # _G. C. Greubel_, Mar 21 2019
%o (GAP) List([3..30], n-> Binomial(2*n, n-3)) # _G. C. Greubel_, Mar 21 2019
%Y Diagonal 7 of triangle A100257.
%Y Column k=1 of A263776.
%Y Cf. A001622.
%Y Cf. binomial(2*n+m, n): A000984 (m = 0), A001700 (m = 1), A001791 (m = 2), A002054 (m = 3), A002694 (m = 4), A003516 (m = 5), A030053 - A030056, A004310 - A004318.
%K nonn,easy,changed
%O 3,2
%A _N. J. A. Sloane_
%E More terms from _Emeric Deutsch_, Feb 18 2004