cp's OEIS Frontend

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

%I A000515 M4874 N2087 #85 Apr 15 2025 08:58:45
%S A000515 1,12,180,2800,44100,698544,11099088,176679360,2815827300,44914183600,
%T A000515 716830370256,11445589052352,182811491808400,2920656969720000,
%U A000515 46670906271240000,745904795339462400,11922821963004219300,190600129650794094000,3047248986392325330000
%N A000515 a(n) = (2n)!(2n+1)!/n!^4, or equally (2n+1)*binomial(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_, 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_{i=0..n} binomial(-1/2, i)*binomial(1/2, i). - Yong Kong (ykong(AT)curagen.com), Dec 26 2000
%C A000515 Right-hand side of Sum_{i=0..n} Sum_{j=0..n} binomial(i+j,j)^2 * binomial(4n-2i-2j,2n-2j).
%D A000515 E. R. Hansen, A Table of Series and Products, Prentice-Hall, Englewood Cliffs, NJ, 1975, p. 96.
%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.
%D A000515 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D A000515 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A000515 T. D. Noe, <a href="/A000515/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.
%H A000515 D. Galakhov, A. Mironov and A. Morozov, <a href="http://arxiv.org/abs/1410.8482">Wall Crossing Invariants: from quantum mechanics to knots</a>, arXiv preprint arXiv:1410.8482 [hep-th], 2014. See Eq. (A.15).
%H A000515 R. K. Guy, <a href="/A005249/a005249_1.pdf">Letter to N. J. A. Sloane, Sep 1986</a>
%H A000515 J. E. Lauer, <a href="/A005249/a005249.pdf">Letter to N. J. A. Sloane, Dec 1980</a>
%H A000515 D. H. Lehmer, Review of A. N. Lowan et al., "Table of the zeros of the Legendre polynomials of order 1-16...", in <a href="http://dx.doi.org/10.1090/S0025-5718-43-99135-5">Math. Tables Aids Computation (MTAC)</a>, 1 (1943-1945), 52-53.
%H A000515 Pedro J. Miana and Natalia Romero, <a href="https://doi.org/10.1016/j.jnt.2010.01.018">Moments of combinatorial and Catalan numbers</a>, Journal of Number Theory, Volume 130, Issue 8, August 2010, Pages 1876-1887. See Omega3. Remark 3 p. 1882.
%H A000515 I. Nemes et al., <a href="http://www.jstor.org/stable/2975078">How to do Monthly problems with your computer</a>, Amer. Math. Monthly, 104 (1997), 505-519.
%H A000515 Yidong Sun and Fei Ma, <a href="http://arxiv.org/abs/1305.2017">Four transformations on the Catalan triangle</a>, arXiv preprint arXiv:1305.2017 [math.CO], 2013 (see Omega_3).
%H A000515 Yidong Sun and Fei Ma, <a href="https://doi.org/10.37236/3701">Some new binomial sums related to the Catalan triangle</a>, Electronic Journal of Combinatorics 21(1) (2014), #P1.33
%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_, 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_, 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
%F A000515 E.g.f.: x*(BesselI(0, 2*x))^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
%F A000515 n^2*a(n) - 4*(2*n-1)*(2*n+1)*a(n-1) = 0. - _R. J. Mathar_, Sep 08 2013
%F A000515 O.g.f.: hypergeom([1/2, 3/2], [1], 16*x). - _Peter Luschny_, Oct 08 2015
%p A000515 with(linalg): for n from 1 to 24 do print(det(hilbert(n))/det(hilbert(n+1))): od;
%t A000515 A000515[n_] := (2*n + 1)*Binomial[2 n, n]^2 (* _Enrique Pérez Herrero_, Mar 31 2010 *)
%t A000515 Table[(2 n + 1) (n + 1)^2 CatalanNumber[n]^2, {n, 0, 18}] (* _Jan Mangaldan_, Sep 23 2021 *)
%o A000515 (Magma) [(2*n+1)*Binomial(2*n,n)^2: n in [0..25]]; // _Vincenzo Librandi_, Oct 08 2015
%o A000515 (PARI) vector(100, n, n--; (2*n+1)*binomial(2*n,n)^2) \\ _Altug Alkan_, Oct 08 2015
%Y A000515 Cf. A002894, A005249, A002457, A000108, A039598, A024492, A000894, A228329, A000515, A228330, A228331, A228332, A228333.
%K A000515 nonn,easy,nice
%O A000515 0,2
%A A000515 _N. J. A. Sloane_