login
The OEIS is supported by the many generous donors to the OEIS Foundation.

 

Logo

Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 59th year, we have over 358,000 sequences, and we’ve crossed 10,300 citations (which often say “discovered thanks to the OEIS”).

Other ways to Give
Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A221954 a(n) = 3^(n-1) * n! * Catalan(n-1). 9
1, 6, 108, 3240, 136080, 7348320, 484989120, 37829151360, 3404623622400, 347271609484800, 39588963481267200, 4988209398639667200, 688372897012274073600, 103255934551841111040000, 16727461397398259988480000, 2910578283147297237995520000, 541367560665397286267166720000, 107190777011748662680899010560000, 22510063172467219162988792217600000 (list; graph; refs; listen; history; text; internal format)
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

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).

Cf. A000108.

Sequence in context: A288148 A010563 A114310 * A167484 A011555 A122722

Adjacent sequences: A221951 A221952 A221953 * A221955 A221956 A221957

KEYWORD

nonn

AUTHOR

N. J. A. Sloane, Feb 03 2013

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified December 2 05:20 EST 2022. Contains 358485 sequences. (Running on oeis4.)