A000515 a(n) = (2n)!(2n+1)!/n!^4, or equally (2n+1)*binomial(2n,n)^2. (Formerly M4874 N2087)

1, 12, 180, 2800, 44100, 698544, 11099088, 176679360, 2815827300, 44914183600, 716830370256, 11445589052352, 182811491808400, 2920656969720000, 46670906271240000, 745904795339462400, 11922821963004219300, 190600129650794094000, 3047248986392325330000

Offset

0,2

Comments

a(n) is also the (n,n)-th entry in the inverse of the n-th Hilbert matrix. - Asher Auel, 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_{i=0..n} binomial(-1/2, i)*binomial(1/2, i). - Yong Kong (ykong(AT)curagen.com), Dec 26 2000

Right-hand side of Sum_{i=0..n} Sum_{j=0..n} binomial(i+j,j)^2 * binomial(4n-2i-2j,2n-2j).

References

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

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.

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

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

Links

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.

D. Galakhov, A. Mironov and A. Morozov, Wall Crossing Invariants: from quantum mechanics to knots, arXiv preprint arXiv:1410.8482 [hep-th], 2014. See Eq. (A.15).

R. K. Guy, Letter to N. J. A. Sloane, Sep 1986

J. E. Lauer, Letter to N. J. A. Sloane, Dec 1980

D. H. Lehmer, Review of A. N. Lowan et al., "Table of the zeros of the Legendre polynomials of order 1-16...", in Math. Tables Aids Computation (MTAC), 1 (1943-1945), 52-53.

Pedro J. Miana and Natalia Romero, Moments of combinatorial and Catalan numbers, Journal of Number Theory, Volume 130, Issue 8, August 2010, Pages 1876-1887. See Omega3. Remark 3 p. 1882.

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

Yidong Sun and Fei Ma, Four transformations on the Catalan triangle, arXiv preprint arXiv:1305.2017 [math.CO], 2013 (see Omega_3).

Yidong Sun and Fei Ma, Some new binomial sums related to the Catalan triangle, Electronic Journal of Combinatorics 21(1) (2014), #P1.33

Formula

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, 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, 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

E.g.f.: x*(BesselI(0, 2x))^2 = x+(2*x^3)/(U(0)-2*x^2); U(k) = (2*x^2)*(2*k+1)+(k+1)^3 - (2*x^2)*(2*k+3)*((k+1)^3)/U(k+1)); (continued fraction). - Sergei N. Gladkovskii, Nov 23 2011

n^2*a(n) - 4*(2*n-1)*(2*n+1)*a(n-1) = 0. - R. J. Mathar, Sep 08 2013

O.g.f.: hypergeom([1/2, 3/2], [1], 16*x). - Peter Luschny, Oct 08 2015

Maple

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

Mathematica

A000515[n_] := (2*n + 1)*Binomial[2 n, n]^2 (* Enrique Pérez Herrero, Mar 31 2010 *)
Table[(2 n + 1) (n + 1)^2 CatalanNumber[n]^2, {n, 0, 18}] (* Jan Mangaldan, Sep 23 2021 *)

Prog

(Magma) [(2*n+1)*Binomial(2*n, n)^2: n in [0..25]]; // Vincenzo Librandi, Oct 08 2015
(PARI) vector(100, n, n--; (2*n+1)*binomial(2*n, n)^2) \\ Altug Alkan, Oct 08 2015

Crossrefs

Cf. A002894, A005249, A002457, A000108, A039598, A024492, A000894, A228329, A000515, A228330, A228331, A228332, A228333.

Sequence in context: A130550 A073975 A069685 * A241710 A318245 A051609

Adjacent sequences: A000512 A000513 A000514 * A000516 A000517 A000518

Keyword

nonn,easy,nice

Author

N. J. A. Sloane

Status

approved