cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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

Original entry on oeis.org

1, 6, 60, 700, 8820, 116424, 1585584, 22084920, 312869700, 4491418360, 65166397296, 953799087696, 14062422446800, 208618354980000, 3111393751416000, 46619049708716400, 701342468412012900
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Comments

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

Examples

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

References

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

Links

Crossrefs

First differences of A082578.
Cf. A000108, A000142, A000217, A000891, A000984, A001700, A002463, A002894, A103371, A132813, A241530, A248845.

Programs

  • Haskell
    a000894 n = a132813 (2 * n) n  -- Reinhard Zumkeller, Apr 04 2014
  • Magma
    [Factorial(2*n)*Factorial(2*n+1) /(Factorial(n+1)* Factorial(n)^3): n in [0..20]]; // Vincenzo Librandi, Oct 25 2011
  • Magma
    A000894:= func< n | Binomial(2*n+2,2)*Catalan(n)^2 >;
[A000894(n): n in [0..40]]; // G. C. Greubel, Mar 12 2025
  • 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 *)
  • PARI
    {a(n) =  binomial( 2*n + 1, n) * binomial( 2*n, n)}; /* Michael Somos, May 28 2014 */
  • SageMath
    def A000894(n): return binomial(2*n+2,2)*catalan_number(n)^2
print([A000894(n) for n in range(41)]) # G. C. Greubel, Mar 12 2025

Formula

From Zerinvary Lajos, Jan 23 2007: (Start)
a(n) = C(2*n+1,n)*C(2*n,n) = A001700(n)*A000984(n).
a(n) = A000984(n)*A000984(n+1)/2, n>=0. (End)
G.f.: (EllipticK(4*sqrt(x)) - EllipticE(4*sqrt(x)))/(4*Pi*x). - Mark van Hoeij, Oct 24 2011
n*(n+1)*a(n) = 4*(2*n-1)*(2*n+1)*a(n-1). - 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) = (n+1) * A000891(n) = A248045(n+1) / A000142(n). - Reinhard Zumkeller, Sep 30 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
a(n) = A000217(n+1)*A000108(n)*A000108(n+1) = A000217(2*n+1)*A000108(n)^2. - G. C. Greubel, Mar 12 2025

A002462 Coefficients of Legendre polynomials.

Original entry on oeis.org

1, 1, 9, 50, 1225, 7938, 106722, 736164, 41409225, 295488050, 4266847442, 31102144164, 914057459042, 6760780022500, 100583849722500, 751920156592200, 90324408810638025, 680714748752836050, 10294760089163261250, 78080479568224402500, 2375208188465386324050
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Comments

Appears to divide A002894(n+1). - Ralf Stephan, Aug 23 2004
Constant term of the Legendre polynomials of even order when they are expressed in terms of the cosine function (see 22.3.13 from Abramowitz & Stegun) with the denominators factored out. Also, constant term of the Tisserand functions of even order for the planar case with the denominators factored out (see Table 1 from Laskar & Boué's paper). Cf. A002463. - Ruperto Corso, Dec 08 2011

References

  • A. Fletcher, J. C. P. Miller, L. Rosenhead and L. J. Comrie, An Index of Mathematical Tables. Vols. 1 and 2, 2nd ed., Blackwell, Oxford and Addison-Wesley, Reading, MA, 1962, Vol. 1, p. 362.
  • G. Prévost, Tables de Fonctions Sphériques. Gauthier-Villars, Paris, 1933, pp. 156-157.
  • 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

Programs

  • Maple
    f:=(n,q)->binomial(2*(n-q),(n-q))*binomial(2*q,q)/(4^n): seq(f(2*m,m)*lcm(seq(denom(2*f(2*m,i)), i=0..m-1), denom(f(2*m,m))), m=0..25); # Ruperto Corso, Dec 08 2011
  • Mathematica
    f[n_, q_] := Binomial[2(n-q), n-q] Binomial[2q, q]/4^n;
a[m_] := f[2m, m] LCM @@ Append[Table[Denominator[2f[2m, i]], {i, 0, m-1}], Denominator[f[2m, m]]];
Table[a[m], {m, 0, 25}] (* Jean-François Alcover, Jan 20 2019, after Ruperto Corso *)

Formula

This is binomial(2*n,n)^2/2^(4*n) multiplied by some power of 2, but the exact power of 2 needed is a bit hard to describe precisely. No doubt Prévost or Fletcher et al., where I saw this sequence 40 years ago, will have the answer. - N. J. A. Sloane, Jun 01 2013

Extensions

Sequence extended by Ruperto Corso, Dec 08 2011
Showing 1-2 of 2 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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