A052714 a(n) = 2^(n-1) * n! * Catalan(n-1) for n > 0 with a(0) = 0.

0, 1, 4, 48, 960, 26880, 967680, 42577920, 2214051840, 132843110400, 9033331507200, 686533194547200, 57668788341964800, 5305528527460761600, 530552852746076160000, 57299708096576225280000, 6646766139202842132480000, 824199001261152424427520000

Offset: 0

Author

encyclopedia(AT)pommard.inria.fr, Jan 25 2000

Keywords

• easy
• nonn

Comments

a(n+1) is the number of square roots of any permutation in S_{8*n} whose disjoint cycle decomposition consists of 2*n cycles of length 4. - Luis Manuel Rivera Martínez, Feb 26 2015

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..200
• INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 670.
• Jesús Leaños, Rutilo Moreno and Luis Manuel Rivera-Martínez, On the number of mth roots of permutations, arXiv:1005.1531 [math.CO], 2010-2011.
• Jesús Leaños, Rutilo Moreno and Luis Manuel Rivera-Martínez, On the number of mth roots of permutations, Australas. J. Combin., Vol. 52 (2012), pp. 41-54 (Theorem 1).

Crossrefs

Sequences of the form m^(n-1)*n!*Catalan(n-1): A001813 (m=1), this sequence (or A144828) (m=2), A221954 (m=3), A052734 (m=4), A221953 (m=5), A221955 (m=6).

Cf. A000108, A221953, A221954.

Programs

Magma

[0] cat [Catalan(n-1)*2^(n-1)*Factorial(n): n in [1..20]]; // Vincenzo Librandi, Mar 11 2013

Maple

spec := [S,{B=Union(Z,C),S=Union(B,C),C=Prod(S,S)},labeled]: seq(combstruct[count](spec,size=n), n=0..20);

Mathematica

Join[{0}, Table[CatalanNumber[n-1] 2^(n-1) n!, {n,  1, 20}]] (* Vincenzo Librandi, Mar 11 2013 *)

PARI

a(n)=if(n<1,0,2^(n-1)*(2*n-2)!/(n-1)!)

Sage

[0]+[2^(n-1)*factorial(n)*catalan_number(n-1) for n in (1..30)] # G. C. Greubel, Apr 02 2021

Formula

E.g.f.: (1- sqrt(1-8*x))/4.

Recurrence: a(1) = 1, 4*(1 - 2*n)*a(n) + a(n+1) = 0.

a(n) = A052701(n)*n!.

a(n) = 8^(n-1)*Gamma(n-1/2)/Pi^(1/2), n>0.

a(n+1) = A090802(2n, n). - Ross La Haye, Oct 18 2005

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

E.g.f. A(x) satisfies differential equation A'(x)=1/(1-4*A(x)). - Vladimir Kruchinin, May 04 2011

G.f.: x/(1-4x/(1-8x/(1-12x/(1-16x/(1-20x/(1-24x/(1-28x/(1-32x/(1-... (continued fraction). - Philippe Deléham, Jan 07 2012

G.f.: 2*x/G(0), where G(k)= 1 + 1/(1 - 2*x*(8*k+4)/(2*x*(8*k+4) - 1 + 16*x*(k+1)/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, May 30 2013

a(0) = 0, a(1) = 1; a(n) = 2 * Sum_{k=1..n-1} binomial(n,k) * a(k) * a(n-k). - Ilya Gutkovskiy, Jul 09 2020

From Amiram Eldar, Jan 08 2022: (Start)

Sum_{n>=1} 1/a(n) = 1 + e^(1/8)*sqrt(Pi)*erf(1/(2*sqrt(2)))/(2*sqrt(2)), where erf is the error function.

Sum_{n>=1} (-1)^(n+1)/a(n) = 1 - e^(-1/8)*sqrt(Pi)*erfi(1/(2*sqrt(2)))/(2*sqrt(2)), where erfi is the imaginary error function. (End)

Extensions

Edited by N. J. A. Sloane, Feb 03 2013