A221955 a(n) = 6^(n-1) * n! * Catalan(n-1).

1, 12, 432, 25920, 2177280, 235146240, 31039303680, 4842131374080, 871583647334400, 177803064056217600, 40539098604817612800, 10215852848414038425600, 2819575386162274605465600, 845872615848682381639680000, 274062727534973091651256320000, 95373829182170635894637199360000, 35479064455767476552805038161920000

Offset: 1

Author

N. J. A. Sloane, Feb 03 2013

Keywords

• nonn

Comments

a(n+1) is the number of square roots of any permutation in S_{24*n} whose disjoint cycle decomposition consists of 2*n cycles of length 12. - Luis Manuel Rivera Martínez, Feb 28 2015

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..200
• W. van der Aalst, J. Buijs and B. van Dongen, Towards Improving the Representational Bias of Process Mining, 2012.
• 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), A052714 (or A144828) (m=2), A221954 (m=3), A052734 (m=4), A221953 (m=5), this sequence (m=6).

Cf. A000108.

Programs

Magma

[Catalan(n-1)*6^(n-1)*Factorial(n): n in [1..20]]; // Vincenzo Librandi, Mar 11 2013

Maple

A221955:= n-> 3*6^(n-2)*n!*binomial(2*n,n)/(2*n-1); seq(A221955(n), n=1..30); # G. C. Greubel, Apr 02 2021

Mathematica

Table[CatalanNumber[n-1] 6^(n-1) n!, {n, 20}] (* Vincenzo Librandi, Mar 11 2013 *)
nxt[{n_,a_}]:={n+1,12a(2n-1)}; NestList[nxt,{1,1},20][[;;,2]] (* Harvey P. Dale, Sep 21 2024 *)

PARI

my(x='x+O('x^22)); Vec(serlaplace((1-sqrt(1-24*x))/12)) \\ Michel Marcus, Mar 04 2015

Sage

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

Formula

a(n) = 12*(2*n-3)*a(n-1) with a(1)=1. - Bruno Berselli, Mar 11 2013

E.g.f.: (1-sqrt(1-24*x))/12. - Luis Manuel Rivera Martínez, Mar 04 2015

a(1) = 1; a(n) = 6 * Sum_{k=1..n-1} binomial(n,k) * a(k) * a(n-k). - Ilya Gutkovskiy, Jul 10 2020

From Amiram Eldar, Jan 08 2022: (Start)

Sum_{n>=1} 1/a(n) = 1 + e^(1/24)*sqrt(Pi)*erf(1/(2*sqrt(6)))/(2*sqrt(6)), where erf is the error function.

Sum_{n>=1} (-1)^(n+1)/a(n) = 1 - e^(-1/24)*sqrt(Pi)*erfi(1/(2*sqrt(6)))/(2*sqrt(6)), where erfi is the imaginary error function. (End)