A052502 - OEIS

%I #95 Aug 26 2024 09:47:08

%S 1,2,40,2240,246400,44844800,12197785600,4635158528000,

%T 2345390215168000,1524503639859200000,1237896955565670400000,

%U 1227993779921145036800000,1461312598106162593792000000,2054605512937264606871552000000

%N Number of permutations sigma of [3n] without fixed points such that sigma^3 = Id.

%C For n >= 1 a(n) is the size of the conjugacy class in the symmetric group S_(3n) consisting of permutations whose cycle decomposition is a product of n disjoint 3-cycles.

%D F. W. J. Olver, Asymptotics and special functions, Academic Press, 1974, pages 336-344.

%H G. C. Greubel, <a href="/A052502/b052502.txt">Table of n, a(n) for n = 0..210</a>

%H INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=27">Encyclopedia of Combinatorial Structures 27</a>

%H F. W. J. Olver et al., <a href="http://dlmf.nist.gov/9.10#E17">NIST Digital Library of Mathematical Functions</a>, eq. 9.10.17.

%F From Ahmed Fares (ahmedfares(AT)my-deja.com), Apr 21 2001: (Start)

%F a(n) = (3*n)!/(3^n * n!).

%F a(n) ~ sqrt(3) * 9^n * (n/e)^(2n). (End)

%F E.g.f.: (every third coefficient of) exp(x^3/3).

%F G.f.: hypergeometric3F0([1/3, 2/3, 1], [], 9*x).

%F D-finite with recurrence a(n) = (3*n-1)*(3*n-2)*a(n-1) for n >= 1, with a(0) = 1.

%F Write the generating function for this sequence in the form A(x) = Sum_{n >= 0} a(n)* x^(2*n+1)/(2*n+1)!. The g.f. A(x) satisfies A'(x)*( 1 - A(x)^2) = 1. _Robert Israel_ remarks that consequently A(x) is a root of z^3 - 3*z + 3*x with A(0) = 0. Cf. A001147, A052504 and A060706. - _Peter Bala_, Jan 02 2015

%F From _Peter Bala_, Feb 27 2024: (Start)

%F u(n) := a(n+1) satisfies the second-order recurrence u(n) = 18*n*u(n-1) + (3*n - 1)^2*(3*n - 2)^2*u(n-2) with u(0) = 2 and u(1) = 40.

%F A second solution to the recurrence is given by v(n) := u(n)*Sum_{k = 0..n} (-1)^k/((3*k + 1)*(3*k + 2)) with v(0) = 1 and v(1) = 18.

%F This leads to the continued fraction expansion (2/3)*log(2) = Sum_{k = 0..n} (-1)^k/((3*k + 1)*(3*k + 2)) = Limit_{n -> oo} v(n)/u(n) = 1/(2 + (1*2)^2/(18 + (4*5)^2/(2*18 + (7*8)^2/(3*18 + (10*11)^2/(4*18 +  ... ))))). (End)

%F From _Gabriel B. Apolinario_, Jul 30 2024: (Start)

%F a(n) = 3 * Integral_{t=0..oo} Ai(t)*t^(3*n) dt, where Ai(t) is the Airy function.

%F a(n) = Integral_{t=-oo..oo} Ai(t)*t^(3*n) dt. (End)

%p spec := [S,{S=Set(Union(Cycle(Z,card=3)))},labeled]: seq(combstruct[count](spec,size=n), n=0..20);

%t Table[(3*n)!/(3^n*n!), {n, 0, 20}] (* _G. C. Greubel_, May 14 2019 *)

%o (PARI) {a(n) = (3*n)!/(3^n*n!)}; \\ _G. C. Greubel_, May 14 2019

%o (Magma) [Factorial(3*n)/(3^n*Factorial(n)): n in [0..20]]; // _G. C. Greubel_, May 14 2019

%o (Sage) [factorial(3*n)/(3^n*factorial(n)) for n in (0..20)] # _G. C. Greubel_, May 14 2019

%o (GAP) List([0..20], n-> Factorial(3*n)/(3^n*Factorial(n))) # _G. C. Greubel_, May 14 2019

%Y Cf. A000142. Row sums of triangle A060063.

%Y First column of array A091752 (also negative of second column).

%Y Equals row sums of A157702. - _Johannes W. Meijer_, Mar 07 2009

%Y _Karol A. Penson_ suggested that the row sums of A060063 coincide with this entry.

%Y Cf. A001147, A052504, A060706, A261317, A261381.

%Y Trisection of column k=3 of A261430.

%K easy,nonn

%O 0,2

%A encyclopedia(AT)pommard.inria.fr, Jan 25 2000

%E Edited by _Wolfdieter Lang_, Feb 13 2004

%E Title improved by _Geoffrey Critzer_, Aug 14 2015