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 A000894 a(n) = (2*n)!*(2*n+1)! /((n+1)! *n!^3). 20

%I

%S 1,6,60,700,8820,116424,1585584,22084920,312869700,4491418360,

%T 65166397296,953799087696,14062422446800,208618354980000,

%U 3111393751416000,46619049708716400,701342468412012900

%N a(n) = (2*n)!*(2*n+1)! /((n+1)! *n!^3).

%C a(n) = (n+1) * A000891(n) = A248045(n+1) / A000142(n). - _Reinhard Zumkeller_, Sep 30 2014

%C This sequence is one half of the odd part of the bisection of A241530. The even part is given in A002894. - _Wolfdieter Lang_, Sep 06 2016

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

%H Vincenzo Librandi, <a href="/A000894/b000894.txt">Table of n, a(n) for n = 0..180</a>

%H 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_1).

%H Yidong Sun and Fei Ma, <a href="http://www.combinatorics.org/ojs/index.php/eljc/article/view/v21i1p33">Some new binomial sums related to the Catalan triangle</a>, Electronic Journal of Combinatorics 21(1) (2014), #P1.33.

%F a(n) = C(2*n+1,n)*C(2*n,n) = A001700(n)*A000984(n) = A000984(n)*A000984(n+1)/2, n>=0. - _Zerinvary Lajos_, Jan 23 2007

%F G.f.: (EllipticK(4*x^(1/2)) - EllipticE(4*x^(1/2)))/(4*x*Pi). - Mark van Hoeij, Oct 24 2011

%F n*(n+1)*a(n) -4*(2*n-1)*(2*n+1)*a(n-1)=0. - _R. J. Mathar_, Sep 08 2013

%F a(n) = A103371(2*n,n) = A132813(2*n,n). - _Reinhard Zumkeller_, Apr 04 2014

%F 0 = a(n)*(+65536*a(n+2) - 23040*a(n+3) + 1400*a(n+4)) + a(n+1)*(-1536*a(n+2) + 1184*a(n+3) - 90*a(n+4)) + a(n+2)*(-24*a(n+2) - 6*a(n+3) + a(n+4)) for all n in Z. - _Michael Somos_, May 28 2014

%F 0 = a(n+1)^3 * (+256*a(n) - 6*a(n+1) + a(n+2)) + a(n) * a(n+1) * a(n+

%F 2) * (-768*a(n) - 20*a(n+1) - 3*a(n+2)) + 90*a(n)^2*a(n+2)^2 for all n in Z. - _Michael Somos_, Sep 17 2014

%F a(n) = A241530(2n+1)/2, n >= 0. - _Wolfdieter Lang_, Sep 06 2016

%F a(n) ~ 2^(4*n+1)/(Pi*n). - _Ilya Gutkovskiy_, Sep 06 2016

%e G.f. = 1 + 6*x + 60*x^2 + 700*x^3 + 8820*x^4 + 116424*x^5 + ...

%p seq(binomial(2*n+1,n)*binomial(2*n,n), n=0..16); # _Zerinvary Lajos_, Jan 23 2007

%t a[ n_] := Binomial[2 n + 1, n] Binomial[2 n, n]; (* _Michael Somos_, May 28 2014 *)

%t a[ n_] := SeriesCoefficient[ (EllipticK[ 16 x] - EllipticE[ 16 x]) / (4 x Pi), {x, 0, n}]; (* _Michael Somos_, May 28 2014 *)

%t Table[(2 n)!*(2 n + 1)!/((n + 1)!*n!^3), {n, 0, 16}] (* _Michael De Vlieger_, Sep 06 2016 *)

%o (MAGMA) [Factorial(2*n)*Factorial(2*n+1) /(Factorial(n+1)* Factorial(n)^3): n in [0..20]]; // _Vincenzo Librandi_, Oct 25 2011

%o a000894 n = a132813 (2 * n) n -- _Reinhard Zumkeller_, Apr 04 2014

%o (PARI) {a(n) = binomial( 2*n + 1, n) * binomial( 2*n, n)}; /* _Michael Somos_, May 28 2014 */

%Y First differences of A082578. Cf. A002463.

%Y Cf. A001700, A000984.

%Y Cf. A000142, A000891, A248045, A002894, A241530.

%K nonn,easy

%O 0,2

%A _N. J. A. Sloane_

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Last modified February 16 21:59 EST 2019. Contains 320200 sequences. (Running on oeis4.)