A221955 - OEIS

%I #54 Sep 21 2024 19:14:26

%S 1,12,432,25920,2177280,235146240,31039303680,4842131374080,

%T 871583647334400,177803064056217600,40539098604817612800,

%U 10215852848414038425600,2819575386162274605465600,845872615848682381639680000,274062727534973091651256320000,95373829182170635894637199360000,35479064455767476552805038161920000

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

%C 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

%H Vincenzo Librandi, <a href="/A221955/b221955.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://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) = 12*(2*n-3)*a(n-1) with a(1)=1. - _Bruno Berselli_, Mar 11 2013

%F E.g.f.: (1-sqrt(1-24*x))/12. - _Luis Manuel Rivera Martínez_, Mar 04 2015

%F a(1) = 1; a(n) = 6 * Sum_{k=1..n-1} binomial(n,k) * a(k) * a(n-k). - _Ilya Gutkovskiy_, Jul 10 2020

%F From _Amiram Eldar_, Jan 08 2022: (Start)

%F 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.

%F 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)

%p 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

%t Table[CatalanNumber[n-1] 6^(n-1) n!, {n, 20}] (* _Vincenzo Librandi_, Mar 11 2013 *)

%t nxt[{n_,a_}]:={n+1,12a(2n-1)}; NestList[nxt,{1,1},20][[;;,2]] (* _Harvey P. Dale_, Sep 21 2024 *)

%o (Magma) [Catalan(n-1)*6^(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(1-24*x))/12)) \\ _Michel Marcus_, Mar 04 2015

%o (Sage) [6^(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), A221954 (m=3), A052734 (m=4), A221953 (m=5), this sequence (m=6).

%Y Cf. A000108.

%K nonn

%O 1,2

%A _N. J. A. Sloane_, Feb 03 2013