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A052714 a(n) = 2^(n-1) * n! * Catalan(n-1) for n > 0 with a(0) = 0.

%I A052714 #69 Sep 08 2022 08:44:59
%S A052714 0,1,4,48,960,26880,967680,42577920,2214051840,132843110400,
%T A052714 9033331507200,686533194547200,57668788341964800,5305528527460761600,
%U A052714 530552852746076160000,57299708096576225280000,6646766139202842132480000,824199001261152424427520000
%N A052714 a(n) = 2^(n-1) * n! * Catalan(n-1) for n > 0 with a(0) = 0.
%C A052714 a(n+1) is the number of square roots of any permutation in S_{8*n} whose disjoint cycle decomposition consists of 2*n cycles of length 4. - _Luis Manuel Rivera Martínez_, Feb 26 2015
%H A052714 Vincenzo Librandi, <a href="/A052714/b052714.txt">Table of n, a(n) for n = 0..200</a>
%H A052714 INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=670">Encyclopedia of Combinatorial Structures 670</a>.
%H A052714 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 A052714 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 A052714 E.g.f.: (1- sqrt(1-8*x))/4.
%F A052714 Recurrence: a(1) = 1, 4*(1 - 2*n)*a(n) + a(n+1) = 0.
%F A052714 a(n) = A052701(n)*n!.
%F A052714 a(n) = 8^(n-1)*Gamma(n-1/2)/Pi^(1/2), n>0.
%F A052714 a(n+1) = A090802(2n, n). - _Ross La Haye_, Oct 18 2005
%F A052714 a(n) = 2^(n-1)*(2*n-2)!/(n-1)! for n>=1.
%F A052714 E.g.f. A(x) satisfies differential equation A'(x)=1/(1-4*A(x)). - _Vladimir Kruchinin_, May 04 2011
%F A052714 G.f.: x/(1-4x/(1-8x/(1-12x/(1-16x/(1-20x/(1-24x/(1-28x/(1-32x/(1-... (continued fraction). - _Philippe Deléham_, Jan 07 2012
%F A052714 G.f.: 2*x/G(0), where G(k)= 1 + 1/(1 - 2*x*(8*k+4)/(2*x*(8*k+4) - 1 + 16*x*(k+1)/G(k+1))); (continued fraction). - _Sergei N. Gladkovskii_, May 30 2013
%F A052714 a(0) = 0, a(1) = 1; a(n) = 2 * Sum_{k=1..n-1} binomial(n,k) * a(k) * a(n-k). - _Ilya Gutkovskiy_, Jul 09 2020
%F A052714 From _Amiram Eldar_, Jan 08 2022: (Start)
%F A052714 Sum_{n>=1} 1/a(n) = 1 + e^(1/8)*sqrt(Pi)*erf(1/(2*sqrt(2)))/(2*sqrt(2)), where erf is the error function.
%F A052714 Sum_{n>=1} (-1)^(n+1)/a(n) = 1 - e^(-1/8)*sqrt(Pi)*erfi(1/(2*sqrt(2)))/(2*sqrt(2)), where erfi is the imaginary error function. (End)
%p A052714 spec := [S,{B=Union(Z,C),S=Union(B,C),C=Prod(S,S)},labeled]: seq(combstruct[count](spec,size=n), n=0..20);
%t A052714 Join[{0}, Table[CatalanNumber[n-1] 2^(n-1) n!, {n,  1, 20}]] (* _Vincenzo Librandi_, Mar 11 2013 *)
%o A052714 (PARI) a(n)=if(n<1,0,2^(n-1)*(2*n-2)!/(n-1)!)
%o A052714 (Magma) [0] cat [Catalan(n-1)*2^(n-1)*Factorial(n): n in [1..20]]; // _Vincenzo Librandi_, Mar 11 2013
%o A052714 (Sage) [0]+[2^(n-1)*factorial(n)*catalan_number(n-1) for n in (1..30)] # _G. C. Greubel_, Apr 02 2021
%Y A052714 Sequences of the form m^(n-1)*n!*Catalan(n-1): A001813 (m=1), this sequence (or A144828) (m=2), A221954 (m=3), A052734 (m=4), A221953 (m=5), A221955 (m=6).
%Y A052714 Cf. A000108, A221953, A221954.
%K A052714 easy,nonn
%O A052714 0,3
%A A052714 encyclopedia(AT)pommard.inria.fr, Jan 25 2000
%E A052714 Edited by _N. J. A. Sloane_, Feb 03 2013