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A221955
a(n) = 6^(n-1) * n! * Catalan(n-1).
7
1, 12, 432, 25920, 2177280, 235146240, 31039303680, 4842131374080, 871583647334400, 177803064056217600, 40539098604817612800, 10215852848414038425600, 2819575386162274605465600, 845872615848682381639680000, 274062727534973091651256320000, 95373829182170635894637199360000, 35479064455767476552805038161920000
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
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
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).
Cf. A000108.
Sequence in context: A129006 A067429 A054913 * A070285 A241593 A041617
KEYWORD
nonn
AUTHOR
N. J. A. Sloane, Feb 03 2013
STATUS
approved