A221953 a(n) = 5^(n-1) * n! * Catalan(n-1).

1, 10, 300, 15000, 1050000, 94500000, 10395000000, 1351350000000, 202702500000000, 34459425000000000, 6547290750000000000, 1374931057500000000000, 316234143225000000000000, 79058535806250000000000000, 21345804667687500000000000000, 6190283353629375000000000000000, 1918987839625106250000000000000000

Offset: 1

Author

N. J. A. Sloane, Feb 03 2013

Keywords

• nonn

Comments

a(n+1) is the number of square roots of any permutation in S_{20*n} whose disjoint cycle decomposition consists of 2*n cycles of length 10. - 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).

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), this sequence (m=5), A221955 (m=6).

Cf. A000108.

Programs

Magma

[Catalan(n-1)*5^(n-1)*Factorial(n): n in [1..20]]; // Vincenzo Librandi, Mar 11 2013

Maple

A221953:= n-> (5^(n-1)*n!/(2*(2*n-1))*binomial(2*n,n); seq(A221953(n), n=1..30); # G. C. Greubel, Apr 02 2021

Mathematica

Table[CatalanNumber[n - 1]  5^(n-1)  n!, {n, 20}] (* Vincenzo Librandi, Mar 11 2013 *)

PARI

my(x='x+O('x^22)); Vec(serlaplace((1-sqrt(1-20*x))/10)) \\ Michel Marcus, Mar 04 2015

Sage

[5^(n-1)*factorial(n)*catalan_number(n-1) for n in (1..30)] # G. C. Greubel, Apr 02 2021

Formula

a(n) = 10*(2*n-3)*a(n-1) with a(1)=1. - Bruno Berselli, Mar 11 2013

E.g.f.: (1 - sqrt(1-20*x))/10. - Luis Manuel Rivera Martínez, Mar 04 2015

a(1) = 1; a(n) = 5 * 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^(1/20)*sqrt(Pi)*erf(1/(2*sqrt(5)))/(2*sqrt(5)), where erf is the error function.

Sum_{n>=1} (-1)^(n+1)/a(n) = 1 - e^(-1/20)*sqrt(Pi)*erfi(1/(2*sqrt(5)))/(2*sqrt(5)), where erfi is the imaginary error function. (End)