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A002696 Binomial coefficients C(2n,n-3).
(Formerly M4532 N1921)
7

%I M4532 N1921 #74 Dec 06 2022 15:49:25

%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="http://www.pmfbl.org/janjic/">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)

%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.

%K nonn,easy

%O 3,2

%A _N. J. A. Sloane_

%E More terms from _Emeric Deutsch_, Feb 18 2004

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Last modified March 28 07:48 EDT 2024. Contains 371235 sequences. (Running on oeis4.)