OFFSET
1,2
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).
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)
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 *)
PROG
(Magma) [Catalan(n-1)*3^(n-1)*Factorial(n): n in [1..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
CROSSREFS
KEYWORD
nonn
AUTHOR
N. J. A. Sloane, Feb 03 2013
STATUS
approved