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A000894 a(n) = (2*n)!*(2*n+1)! /((n+1)! *n!^3). 19
1, 6, 60, 700, 8820, 116424, 1585584, 22084920, 312869700, 4491418360, 65166397296, 953799087696, 14062422446800, 208618354980000, 3111393751416000, 46619049708716400, 701342468412012900 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

a(n) = (n+1) * A000891(n) = A248045(n+1) / A000142(n). - Reinhard Zumkeller, Sep 30 2014

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

REFERENCES

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

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..180

Yidong Sun and Fei Ma, Four transformations on the Catalan triangle, arXiv preprint arXiv:1305.2017 [math.CO], 2013 (see Omega_1).

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) = 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

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

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

a(n) = A103371(2*n,n) = A132813(2*n,n). - Reinhard Zumkeller, Apr 04 2014

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

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

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

a(n) = A241530(2n+1)/2, n >= 0. - Wolfdieter Lang, Sep 06 2016

a(n) ~ 2^(4*n+1)/(Pi*n). - Ilya Gutkovskiy, Sep 06 2016

EXAMPLE

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

MAPLE

seq(binomial(2*n+1, n)*binomial(2*n, n), n=0..16); # Zerinvary Lajos, Jan 23 2007

MATHEMATICA

a[ n_] := Binomial[2 n + 1, n] Binomial[2 n, n]; (* Michael Somos, May 28 2014 *)

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

Table[(2 n)!*(2 n + 1)!/((n + 1)!*n!^3), {n, 0, 16}] (* Michael De Vlieger, Sep 06 2016 *)

PROG

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

(Haskell)

a000894 n = a132813 (2 * n) n  -- Reinhard Zumkeller, Apr 04 2014

(PARI) {a(n) =  binomial( 2*n + 1, n) * binomial( 2*n, n)}; /* Michael Somos, May 28 2014 */

CROSSREFS

First differences of A082578. Cf. A002463.

Cf. A001700, A000984.

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

Sequence in context: A232969 A232246 A086984 * A112117 A065944 A126779

Adjacent sequences:  A000891 A000892 A000893 * A000895 A000896 A000897

KEYWORD

nonn,easy

AUTHOR

N. J. A. Sloane

STATUS

approved

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Last modified October 19 13:38 EDT 2018. Contains 316361 sequences. (Running on oeis4.)