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%I #88 Mar 26 2023 10:29:33
%S 1,4,24,160,1120,8064,59136,439296,3294720,24893440,189190144,
%T 1444724736,11076222976,85201715200,657270374400,5082890895360,
%U 39392404439040,305870434467840,2378992268083200,18531097667174400
%N a(n) = 2^n * (2*n)! / (n!)^2.
%C Number of lattice paths from (0,0) to (n,n) using steps (0,1), and two kinds of steps (1,0). - _Joerg Arndt_, Jul 01 2011
%C The convolution square root of this sequence is A004981. - _T. D. Noe_, Jun 11 2002
%C Also main diagonal of array: T(i,1)=2^(i-1), T(1,j)=1, T(i,j) = T(i,j-1) + 2*T(i-1,j). - _Benoit Cloitre_, Feb 26 2003
%C The Hankel transform (see A001906 for definition) of this sequence with interpolated zeros(1, 0, 4, 0, 24, 0, 160, 0, 1120, ...) = is A036442: 1, 4, 32, 512, 16384, ... . - _Philippe Deléham_, Jul 03 2005
%C The Hankel transform of this sequence gives A103488. - _Philippe Deléham_, Dec 02 2007
%C Equals the central column of the triangle A038207. - _Zerinvary Lajos_, Dec 08 2007
%H Harry J. Smith, <a href="/A059304/b059304.txt">Table of n, a(n) for n = 0..200</a>
%H Paul Barry and Arnauld Mesinga Mwafise, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL21/Barry/barry362.html">Classical and Semi-Classical Orthogonal Polynomials Defined by Riordan Arrays, and Their Moment Sequences</a>, Journal of Integer Sequences, Vol. 21 (2018), Article 18.1.5.
%H Hacène Belbachir and Abdelghani Mehdaoui, <a href="https://doi.org/10.2989/16073606.2020.1729269">Recurrence relation associated with the sums of square binomial coefficients</a>, Quaestiones Mathematicae (2021) Vol. 44, Issue 5, 615-624.
%H Hacène Belbachir, Abdelghani Mehdaoui, and László Szalay, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL22/Szalay/szalay42.html">Diagonal Sums in the Pascal Pyramid, II: Applications</a>, J. Int. Seq., Vol. 22 (2019), Article 19.3.5.
%H H. J. Brothers, <a href="http://www.brotherstechnology.com/docs/Pascal's_Prism_(supplement).pdf">Pascal's Prism: Supplementary Material</a>.
%H Tony D. Noe, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL9/Noe/noe35.html">On the Divisibility of Generalized Central Trinomial Coefficients</a>, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.7.
%H Michael Z. Spivey and Laura L. Steil, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL9/Spivey/spivey7.html">The k-Binomial Transforms and the Hankel Transform</a>, Journal of Integer Sequences, Vol. 9 (2006), Article 06.1.1.
%F a(n) = C(2*n,n) * 2^n.
%F D-finite with recurrence a(n) = a(n-1)*(8-4/n).
%F a(n) = A000079(n)*A000984(n)
%F G.f.: 1/sqrt(1-8*x) - _T. D. Noe_, Jun 11 2002
%F E.g.f.: exp(4*x)*BesselI(0, 4*x). - _Vladeta Jovovic_, Aug 20 2003
%F a(n) = A038207(n,n). - _Joerg Arndt_, Jul 01 2011
%F G.f.: G(0)/2, where G(k) = 1 + 1/(1 - 4*x*(2*k+1)/(4*x*(2*k+1) + (k+1)/G(k+1))); (continued fraction). - _Sergei N. Gladkovskii_, May 24 2013
%F E.g.f.: E(0)/2, where E(k) = 1 + 1/(1 - 4*x/(4*x + (k+1)^2/(2*k+1)/E(k+1))); (continued fraction). - _Sergei N. Gladkovskii_, Jun 01 2013
%F G.f.: Q(0)/(1+2*sqrt(x)), where Q(k) = 1 + 2*sqrt(x)/(1 - 2*sqrt(x)*(2*k+1)/(2*sqrt(x)*(2*k+1) + (k+1)/Q(k+1) )); (continued fraction). - _Sergei N. Gladkovskii_, Aug 09 2013
%F O.g.f.: hypergeom([1/2], [], 8*x). - _Peter Luschny_, Oct 08 2015
%F a(n) = Sum_{k = 0..2*n} (-1)^(n+k)*binomial(2*n,k)*binomial(3*n - 2*k,n)*binomial(n+k,n). - _Peter Bala_, Aug 04 2016
%F a(n) ~ 8^n/sqrt(Pi*n). - _Ilya Gutkovskiy_, Aug 04 2016
%F From _Amiram Eldar_, Jul 21 2020: (Start)
%F Sum_{n>=0} 1/a(n) = 8/7 + 8*sqrt(7)*arcsin(1/sqrt(8))/49.
%F Sum_{n>=0} (-1)^n/a(n) = (8/27)*(3 - arcsinh(1/sqrt(8))). (End)
%F a(n) = Sum_{k = n..2*n} binomial(2*n,k)*binomial(k,n). In general, for m >= 1, Sum_{k = n..m*n} binomial(m*n,k)*binomial(k,n) = 2^((m-1)*n)*binomial(m*n,n). - _Peter Bala_, Mar 25 2023
%p seq(binomial(2*n,n)*2^n,n=0..19); # _Zerinvary Lajos_, Dec 08 2007
%t Table[2^n Binomial[2n,n],{n,0,30}] (* _Harvey P. Dale_, Dec 16 2014 *)
%o (PARI) {a(n)=if(n<0, 0, 2^n*(2*n)!/n!^2)} /* _Michael Somos_, Jan 31 2007 */
%o (PARI) { for (n = 0, 200, write("b059304.txt", n, " ", 2^n * (2*n)! / n!^2); ) } \\ _Harry J. Smith_, Jun 25 2009
%o (PARI) /* as lattice paths: same as in A092566 but use */
%o steps=[[1, 0], [1, 0], [0, 1]]; /* note the double [1, 0] */
%o /* _Joerg Arndt_, Jul 01 2011 */
%o (Magma) [2^n*Factorial(2*n)/Factorial(n)^2: n in [0..25]]; // _Vincenzo Librandi_, Oct 08 2015
%Y Diagonal of A013609.
%Y Cf. A038207.
%Y Column k=0 of A067001.
%K nonn,easy
%O 0,2
%A _Henry Bottomley_, Jan 25 2001
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