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%I #59 Sep 08 2022 08:46:04
%S 1,10,300,15000,1050000,94500000,10395000000,1351350000000,
%T 202702500000000,34459425000000000,6547290750000000000,
%U 1374931057500000000000,316234143225000000000000,79058535806250000000000000,21345804667687500000000000000,6190283353629375000000000000000,1918987839625106250000000000000000
%N a(n) = 5^(n-1) * n! * Catalan(n-1).
%C 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
%H Vincenzo Librandi, <a href="/A221953/b221953.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) = 10*(2*n-3)*a(n-1) with a(1)=1. - _Bruno Berselli_, Mar 11 2013
%F E.g.f.: (1 - sqrt(1-20*x))/10. - _Luis Manuel Rivera Martínez_, Mar 04 2015
%F a(1) = 1; a(n) = 5 * 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^(1/20)*sqrt(Pi)*erf(1/(2*sqrt(5)))/(2*sqrt(5)), where erf is the error function.
%F 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)
%p 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
%t Table[CatalanNumber[n - 1] 5^(n-1) n!, {n, 20}] (* _Vincenzo Librandi_, Mar 11 2013 *)
%o (Magma) [Catalan(n-1)*5^(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(1-20*x))/10)) \\ _Michel Marcus_, Mar 04 2015
%o (Sage) [5^(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), this sequence (m=5), A221955 (m=6).
%Y Cf. A000108.
%K nonn
%O 1,2
%A _N. J. A. Sloane_, Feb 03 2013
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