OFFSET
0,2
COMMENTS
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.
REFERENCES
F. W. J. Olver, Asymptotics and special functions, Academic Press, 1974, pages 336-344.
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..210
Bishal Deb and Alan D. Sokal, Higher-order Stirling cycle and subset triangles: Total positivity, continued fractions and real-rootedness, arXiv:2507.18959 [math.CO], 2025. See p. 7.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 27.
F. W. J. Olver et al., NIST Digital Library of Mathematical Functions, eq. 9.10.17.
FORMULA
From Ahmed Fares (ahmedfares(AT)my-deja.com), Apr 21 2001: (Start)
a(n) = (3*n)!/(3^n * n!).
a(n) ~ sqrt(3) * 9^n * (n/e)^(2n). (End)
E.g.f.: (every third coefficient of) exp(x^3/3).
G.f.: hypergeometric3F0([1/3, 2/3, 1], [], 9*x).
D-finite with recurrence a(n) = (3*n-1)*(3*n-2)*a(n-1) for n >= 1, with a(0) = 1.
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
From Peter Bala, Feb 27 2024: (Start)
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.
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.
This leads to the continued fraction expansion (2/3)*log(2) = Sum_{k >= 0} (-1)^k/((3*k + 1)*(3*k + 2)) = lim_{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)
From Gabriel B. Apolinario, Jul 30 2024: (Start)
a(n) = 3 * Integral_{t=0..oo} Ai(t)*t^(3*n) dt, where Ai(t) is the Airy function.
a(n) = Integral_{t=-oo..oo} Ai(t)*t^(3*n) dt. (End)
With A(x) = Sum_{n >= 0} a(n)*x^(2*n+1)/(2*n+1)! we have sqrt(A'(x)) = 1 + x^2/2! + 17*x^4/4! + 865*x^6/6! + ... is the e.g.f. of A281428. - Peter Bala, Jan 22 2026
EXAMPLE
A(x) = 1 + 2*x^2/2! + 40*x^4/4! + 2240*x^6/6! + 246400*x^8/8! + 44844800*x^10/10! + 12197785600*x^12/12! 4635158528000, 2345390215168000, 1524503639859200000
MAPLE
spec := [S, {S=Set(Union(Cycle(Z, card=3)))}, labeled]: seq(combstruct[count](spec, size=n), n=0..20);
MATHEMATICA
Table[(3*n)!/(3^n*n!), {n, 0, 20}] (* G. C. Greubel, May 14 2019 *)
PROG
(PARI) {a(n) = (3*n)!/(3^n*n!)}; \\ G. C. Greubel, May 14 2019
(Magma) [Factorial(3*n)/(3^n*Factorial(n)): n in [0..20]]; // G. C. Greubel, May 14 2019
(SageMath) [factorial(3*n)/(3^n*factorial(n)) for n in (0..20)] # G. C. Greubel, May 14 2019
(GAP) List([0..20], n-> Factorial(3*n)/(3^n*Factorial(n))); # G. C. Greubel, May 14 2019
CROSSREFS
First column of array A091752 (also negative of second column).
Equals row sums of A157702. - Johannes W. Meijer, Mar 07 2009
Karol A. Penson suggested that the row sums of A060063 coincide with this entry.
Trisection of column k=3 of A261430.
KEYWORD
easy,nonn
AUTHOR
INRIA Encyclopedia of Combinatorial Structures, Jan 25 2000
EXTENSIONS
Edited by Wolfdieter Lang, Feb 13 2004
Title improved by Geoffrey Critzer, Aug 14 2015
STATUS
approved