%I A000515 M4874 N2087
%S A000515 1,12,180,2800,44100,698544,11099088,176679360,2815827300,
%T A000515 44914183600,716830370256,11445589052352,182811491808400,
%U A000515 2920656969720000,46670906271240000,745904795339462400
%N A000515 (2n)!(2n+1)!/n!^4, or equally (2n+1)C(2n,n)^2.
%C A000515 a(n) is also the (n,n)-th entry in the inverse of the n-th Hilbert matrix. 
               - Asher Auel (asher.auel(AT)reed.edu), May 20 2001
%C A000515 a(n) is also the ratio of the determinants of the n-th Hilbert matrix 
               to the (n+1)-th Hilbert matrix (see A005249), for n>0. Thus the determinant 
               of the inverse of the n-th Hilbert matrix is the product of a(i) 
               for i from 1 to n. (Claimed by Jud McCranie without proof, Jul 17 
               2000)
%C A000515 a(n) is the right side of the binomial sum: 2^(4*n) * sum(binomial(-1/
               2, i)*binomial(1/2, i), i=0..n) - Yong Kong (ykong(AT)curagen.com), 
               Dec 26 2000
%C A000515 Right-hand side of Sum[i=0..n, Sum[j=0..n, C(i+j,j)^2 * C(4n-2i-2j,2n-2j)]].
%D A000515 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%D A000515 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 
               (includes this sequence).
%D A000515 E. R. Hansen, A Table of Series and Products, Prentice-Hall, Englewood 
               Cliffs, NJ, 1975, p. 96.
%D A000515 D. H. Lehmer, Review of A. N. Lowan et al., "Table of the zeros of the 
               Legendre polynomials of order 1-16...", Math. Comp., 1 (1943-1945), 
               52-53.
%D A000515 I. Nemes et al., How to do Monthly problems with your computer, Amer. 
               Math. Monthly, 104 (1997), 505-519.
%D A000515 A. P. Prudnikov, Yu. A. Brychkov and O.I. Marichev, "Integrals and Series", 
               Volume 1: "Elementary Functions", Chapter 4: "Finite Sums", New York, 
               Gordon and Breach Science Publishers, 1986-1992.
%H A000515 T. D. Noe, <a href="b000515.txt">Table of n, a(n) for n=0..100</a>
%H A000515 G. E. Andrews and P. Paule, <a href="http://www.risc.uni-linz.ac.at/research/
               combinat/risc/publications/#ppaule">Some questions concerning computer-generated 
               proofs of a binomial double-sum identity</a>, J. Symbolic Computation 
               11(1994), 1-7.
%F A000515 a(n) ~ 2*pi^-1*2^(4*n) - Joe Keane (jgk(AT)jgk.org), Jun 07 2002
%F A000515 O.g.f.: 2/Pi*EllipticE(4*sqrt(x))/(1-16*x). - Vladeta Jovovic (vladeta(AT)eunet.rs), 
               Jun 15 2005
%F A000515 E.g.f.: Sum[n>=0, a(n)*x^(2n)/(2n)! = BesselI(0, 2*x)*(BesselI(0, 2*x)+4*x*BesselI(1, 
               2*x)). - Vladeta Jovovic (vladeta(AT)eunet.rs), Jun 15 2005
%F A000515 E.g.f. Sum_{n>=0} a(n)*x^(2n+1)/(2n+1)! = BesselI(0, 2x)^2*x . - Michael 
               Somos Jun 22 2005
%p A000515 with(linalg): for n from 1 to 24 do print(det(hilbert(n))/det(hilbert(n+1))): 
               od;
%Y A000515 Cf. A002894, A005249, A002457.
%Y A000515 Sequence in context: A130550 A073975 A069685 this_sequence A051609 A001814 
               A013924
%Y A000515 Adjacent sequences: A000512 A000513 A000514 this_sequence A000516 A000517 
               A000518
%K A000515 nonn,easy,nice
%O A000515 0,2
%A A000515 N. J. A. Sloane (njas(AT)research.att.com).