%I #62 Sep 08 2022 08:46:04
%S 1,6,108,3240,136080,7348320,484989120,37829151360,3404623622400,
%T 347271609484800,39588963481267200,4988209398639667200,
%U 688372897012274073600,103255934551841111040000,16727461397398259988480000,2910578283147297237995520000,541367560665397286267166720000,107190777011748662680899010560000,22510063172467219162988792217600000
%N a(n) = 3^(n-1) * n! * Catalan(n-1).
%C 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
%H Vincenzo Librandi, <a href="/A221954/b221954.txt">Table of n, a(n) for n = 1..200</a>
%H W. van der Aalst, J. Buijs and B. van Dongen, <a href="https://hal.inria.fr/hal-01515548">Towards Improving the Representational Bias of Process Mining</a>, 2012.
%H Jesús Leaños, Rutilo Moreno and Luis Manuel Rivera-Martínez, <a href="http://arxiv.org/abs/1005.1531">On the number of mth roots of permutations</a>, arXiv:1005.1531 [math.CO], 2010-2011.
%H Jesús Leaños, Rutilo Moreno and Luis Manuel Rivera-Martínez, <a href="http://ajc.maths.uq.edu.au/pdf/52/ajc_v52_p041.pdf">On the number of mth roots of permutations</a>, Australas. J. Combin., Vol. 52 (2012), pp. 41-54 (Theorem 1).
%F a(n) = 6*(2*n-3)*a(n-1) with a(1)=1. - _Bruno Berselli_, Mar 11 2013
%F E.g.f.: (1 - sqrt(1-12*x))/6. - _Luis Manuel Rivera Martínez_, Mar 04 2015
%F a(n) = 12^(n-1) * Gamma(n - 1/2) / sqrt(Pi). - _Daniel Suteu_, Jan 06 2017
%F a(1) = 1; a(n) = 3 * Sum_{k=1..n-1} binomial(n,k) * a(k) * a(n-k). - _Ilya Gutkovskiy_, Jul 09 2020
%F From _Amiram Eldar_, Jan 08 2022: (Start)
%F 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.
%F 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)
%p 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
%t Table[CatalanNumber[n-1] 3^(n-1) n!, {n, 20}] (* _Vincenzo Librandi_, Mar 11 2013 *)
%o (Magma) [Catalan(n-1)*3^(n-1)*Factorial(n): n in [1..20]]; // _Vincenzo Librandi_, Mar 11 2013
%o (PARI) my(x='x+O('x^22)); Vec(serlaplace((1-sqrt(1-12*x))/6)) \\ _Michel Marcus_, Mar 04 2015
%o (Sage) [3^(n-1)*factorial(n)*catalan_number(n-1) for n in (1..30)] # _G. C. Greubel_, Apr 02 2021
%Y 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).
%Y Cf. A000108.
%K nonn
%O 1,2
%A _N. J. A. Sloane_, Feb 03 2013