0,2

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

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)

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

Right-hand side of Sum[i=0..n, Sum[j=0..n, C(i+j,j)^2 * C(4n-2i-2j,2n-2j)]].

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

E. R. Hansen, A Table of Series and Products, Prentice-Hall, Englewood Cliffs, NJ, 1975, p. 96.

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.

I. Nemes et al., How to do Monthly problems with your computer, Amer. Math. Monthly, 104 (1997), 505-519.

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.

T. D. Noe, Table of n, a(n) for n=0..100

G. E. Andrews and P. Paule, Some questions concerning computer-generated proofs of a binomial double-sum identity, J. Symbolic Computation 11(1994), 1-7.

a(n) ~ 2*pi^-1*2^(4*n) - Joe Keane (jgk(AT)jgk.org), Jun 07 2002

O.g.f.: 2/Pi*EllipticE(4*sqrt(x))/(1-16*x). - Vladeta Jovovic (vladeta(AT)eunet.rs), Jun 15 2005

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

E.g.f. Sum_{n>=0} a(n)*x^(2n+1)/(2n+1)! = BesselI(0, 2x)^2*x . - Michael Somos Jun 22 2005

with(linalg): for n from 1 to 24 do print(det(hilbert(n))/det(hilbert(n+1))): od;

Cf. A002894, A005249, A002457.

Sequence in context: A130550 A073975 A069685 this_sequence A051609 A001814 A013924

Adjacent sequences: A000512 A000513 A000514 this_sequence A000516 A000517 A000518

nonn,easy,nice

N. J. A. Sloane (njas(AT)research.att.com).