A221954 a(n) = 3^(n-1) * n! * Catalan(n-1).

1, 6, 108, 3240, 136080, 7348320, 484989120, 37829151360, 3404623622400, 347271609484800, 39588963481267200, 4988209398639667200, 688372897012274073600, 103255934551841111040000, 16727461397398259988480000, 2910578283147297237995520000, 541367560665397286267166720000, 107190777011748662680899010560000, 22510063172467219162988792217600000

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_{12*n} whose disjoint cycle decomposition consists of 2*n cycles of length 6. - Luis Manuel Rivera Martínez, Feb 26 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, 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), A052714 (or A144828) (m=2), this sequence (m=3), A052734 (m=4), A221953 (m=5), A221955 (m=6).

Cf. A000108.

Programs

Magma

[Catalan(n-1)*3^(n-1)*Factorial(n): n in [1..20]]; // Vincenzo Librandi, Mar 11 2013

Maple

A221954:= n-> (3^(n-1)*n!/(2*(2*n-1))*binomial(2*n,n); seq(A221954(n), n=1..30); # G. C. Greubel, Apr 02 2021

Mathematica

Table[CatalanNumber[n-1] 3^(n-1) n!, {n, 20}] (* Vincenzo Librandi, Mar 11 2013 *)

PARI

my(x='x+O('x^22)); Vec(serlaplace((1-sqrt(1-12*x))/6)) \\ Michel Marcus, Mar 04 2015

Sage

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

Formula

a(n) = 6*(2*n-3)*a(n-1) with a(1)=1. - Bruno Berselli, Mar 11 2013

E.g.f.: (1 - sqrt(1-12*x))/6. - Luis Manuel Rivera Martínez, Mar 04 2015

a(n) = 12^(n-1) * Gamma(n - 1/2) / sqrt(Pi). - Daniel Suteu, Jan 06 2017

a(1) = 1; a(n) = 3 * 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^(1/12)*sqrt(Pi)*erf(1/(2*sqrt(3)))/(2*sqrt(3)), where erf is the error function.

Sum_{n>=1} (-1)^(n+1)/a(n) = 1 - e^(-1/12)*sqrt(Pi)*erfi(1/(2*sqrt(3)))/(2*sqrt(3)), where erfi is the imaginary error function. (End)