This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A060593 #62 Jul 02 2025 16:02:01
%S A060593 1,1,8,180,8064,604800,68428800,10897286400,2324754432000,
%T A060593 640237370572800,221172909834240000,93666727314800640000,
%U A060593 47726800133326110720000,28806532937614688256000000,20325889640780924033433600000,16578303738261941164769280000000
%N A060593 a(n) is the number of ways that a cycle of length 2n+1 in the symmetric group S_(2n+1) can be decomposed as the product of two cycles of length 2n+1.
%C A060593 The sequence deals only with S_m for odd m because for even m the number of representations of an m-cycle in S_m as a product of two m-cycles is zero.
%C A060593 a(n) = product of first 2n-1 numbers divided by their sum. E.g., a(3) = (1*2*3*4*5)/(1+2+3+4+5) = 120/15 = 8. - _Amarnath Murthy_, Jun 03 2004
%C A060593 a(n) is also the number of permutations in Sym(2n) whose "cycle graph" (or "breakpoint graph") contains exactly one alternating cycle, for n>=1 (see Doignon and Labarre). - _Anthony Labarre_, Jun 19 2007
%H A060593 Harry J. Smith, <a href="/A060593/b060593.txt">Table of n, a(n) for n = 0..100</a>
%H A060593 G. Boccara, <a href="http://dx.doi.org/10.1016/0012-365X(80)90001-1">Nombre de représentations d'une permutation comme produit de deux cycles de longueurs données</a>, Discrete Math., Vol. 29, No. 2 (1980), pp. 105-134.
%H A060593 Jean-Paul Doignon and Anthony Labarre, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL10/Doignon/doignon77.html">On Hultman Numbers</a>, J. Integer Seq., Vol. 10 (2007), Article 07.6.2, 13 pages.
%H A060593 Ivan V. Morozov, <a href="https://arxiv.org/abs/2505.16201">On Quotients of a More General Theorem of Wilson</a>, arXiv:2505.16201 [math.NT], 2025. See Z p. 9. Z(2*n+1)=1+a(n), Z(2n)=1.
%H A060593 Karol A. Penson and J.-M. Sixdeniers, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL4/SIXDENIERS/Catalan.html">Integral Representations of Catalan and Related Numbers</a>, J. Integer Sequences, Vol. 4 (2001), Article 01.2.5.
%H A060593 Karol A. Penson and Allan I. Solomon, <a href="https://arxiv.org/abs/quant-ph/0111151">Coherent states from combinatorial sequences</a>, arXiv:quant-ph/0111151, 2001.
%F A060593 a(n) = (2n)! / (n+1).
%F A060593 Integral representation as n-th moment of a positive function on positive half-axis, in Maple notation: a(n)=int(x^n*(exp(-sqrt(x))/sqrt(x)+Ei(-sqrt(x))), x=0..infinity), n=0, 1, 2, ..., where Ei(y) is the exponential integral. This representation is unique. - _Karol A. Penson_, Aug 27 2001
%F A060593 a(n) = n!^2*binomial(2*n,n)/(n+1). - _Zerinvary Lajos_, Jun 06 2006
%F A060593 a(n) = A090586(2*n + 1). - _Gregory Gerard Wojnar_, Jun 10 2021
%F A060593 From _Amiram Eldar_, Feb 08 2022: (Start)
%F A060593 Sum_{n>=0} 1/a(n) = cosh(1) + sinh(1)/2.
%F A060593 Sum_{n>=0} (-1)^n/a(n) = cos(1) - sin(1)/2. (End)
%F A060593 From _Wolfdieter Lang_, Feb 02 2024: (Start)
%F A060593 O.g.f.: hypergeometric([1,1,1,1/2],[2],4*x).
%F A060593 E.g.f.: hypergeometric([1,1,1/2],[2],4*x). (End)
%F A060593 a(n) = A177267(2n+1,n). - _Alois P. Heinz_, Feb 16 2024
%F A060593 D-finite with recurrence (n+1)*a(n) -2*n^2*(2*n-1)*a(n-1)=0. - _R. J. Mathar_, May 26 2025
%e A060593 a(1) = 1 because in S_3 the only way to write the cycle (123) as a product of two 3-cycles is: (123) = (132)(132).
%p A060593 for n from 0 to 25 do printf(`%d,`,(2*n)!/(n+1)) od:
%t A060593 Table[(2*n)!/(n + 1), {n, 0, 13}] (* _Amiram Eldar_, Feb 08 2022 *)
%o A060593 (PARI) a(n) = (2*n)! / (n + 1); \\ _Harry J. Smith_, Jul 07 2009
%Y A060593 Cf. A090586, A177267.
%K A060593 nonn,easy
%O A060593 0,3
%A A060593 Dan Fux (dan.fux(AT)OpenGaia.com or danfux(AT)OpenGaia.com), Apr 12 2001
%E A060593 More terms from _James Sellers_, Apr 13 2001