OFFSET
1,2
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).
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)
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 *)
PROG
(Magma) [Catalan(n-1)*6^(n-1)*Factorial(n): n in [1..20]]; // Vincenzo Librandi, Mar 11 2013
(PARI) my(x='x+O('x^22)); Vec(serlaplace((1-sqrt(1-24*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
CROSSREFS
KEYWORD
nonn
AUTHOR
N. J. A. Sloane, Feb 03 2013
STATUS
approved