A097801 a(n) = (2*n)!/(n!*2^(n-1)).

2, 2, 6, 30, 210, 1890, 20790, 270270, 4054050, 68918850, 1309458150, 27498621150, 632468286450, 15811707161250, 426916093353750, 12380566707258750, 383797567925021250, 12665319741525701250, 443286190953399543750, 16401589065275783118750, 639661973545755541631250

Offset: 0

Author

N. J. A. Sloane, Sep 21 2004

Keywords

• nonn

Comments

Right-hand edge of triangle in A097749.

Also, the number of ways to paint the 2*n cells of dimension n - 1 that bound a regular convex n-cube polytope using exactly 2n colors where n > 0 is the dimension of Euclidean space. - Frank M Jackson, Aug 13 2018

Links

• Muniru A Asiru, Table of n, a(n) for n = 0..150
• Guo-Niu Han, Enumeration of Standard Puzzles, 2011. [Cached copy]
• Guo-Niu Han, Enumeration of Standard Puzzles, arXiv:2006.14070 [math.CO], 2020.

Crossrefs

Cf. A001147, A097749.

Programs

GAP

List([0..20],n->Factorial(2*n)/(Factorial(n)*2^(n-1))); # Muniru A Asiru, Aug 21 2018

Magma

[Factorial(2*n)/(Factorial(n)*2^(n-1)): n in [0..20]]; // Vincenzo Librandi, Aug 21 2018

Maple

a:= proc(n) a(n):= `if`(n=0, 2, a(n-1)*(2*n-1)) end:
seq(a(n), n=0..25);  # Alois P. Heinz, May 27 2013

Mathematica

FoldList[Times, 2, Range[1, 39, 2]] (* Arkadiusz Wesolowski, May 08 2012 *)
2(2*Range[0,20]-1)!! (* Harvey P. Dale, Apr 22 2013 *)
CoefficientList[Series[2/Sqrt[1-2*x], {x, 0, 45}], x]*Table[k !, {k, 0, 45}] (* Stefano Spezia, Sep 04 2018 *)

PARI

x='x+O('x^30); Vec(serlaplace(2*(1-2*x)^(-1/2))) \\ Altug Alkan, Sep 05 2018

Formula

a(n) = 2*A001147(n).

a(n) = 2*(2*n-1)!!. - Johannes W. Meijer, Nov 12 2009

E.g.f.: 2/sqrt(1 - 2*x). - Sergei N. Gladkovskii, Jul 06 2012

G.f.: G(0), where G(k)= 1 + 1/(1 - x*(2*k + 1)/(x*(2*k + 1) + 1/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, May 26 2013

a(n) = 2 * Product_{i=1..n} denominator(i!/(2*i - 1)). - Wesley Ivan Hurt, Oct 12 2013

D-finite with recurrence: a(n) + (-2*n+1)*a(n-1)=0. - R. J. Mathar, Jan 20 2020

From Amiram Eldar, Feb 24 2021: (Start)

Sum_{n>=0} 1/a(n) = 1/2 + sqrt(e*Pi/2)*erf(1/sqrt(2))/2, where erf(x) is the error function.

Sum_{n>=0} (-1)^n/a(n) = 1/2 - sqrt(Pi/(2*e))*erfi(1/sqrt(2))/2, where erfi(x) is the imaginary error function. (End)