A000442 a(n) = (n!)^3.

1, 1, 8, 216, 13824, 1728000, 373248000, 128024064000, 65548320768000, 47784725839872000, 47784725839872000000, 63601470092869632000000, 109903340320478724096000000, 241457638684091756838912000000, 662559760549147780765974528000000, 2236139191853373760085164032000000000

Offset: 0

Author

R. Muller

Keywords

• nonn
• easy

Comments

Permanent of upper right n X n corner of multiplication table (A003991). - Marc LeBrun, Dec 11 2003

a(n) is the number of set partitions of {1, 2, ..., 4n - 1, 4n} into blocks of size 4 in which the entries of each block mod 4 are distinct. For example, a(2) = 8 counts 1234-5678, 1678-2345, 1278-3456, 1346-2578, 1238-4567, 1467-2358, 1247-3568, 1368-2457. - David Callan, Mar 30 2007

a(n) is also the determinant of the symmetric n X n matrix M defined by M(i, j) = sigma_3(gcd(i, j)) for 1 <= i,j <= n, and n > 0, where sigma_3 is A001158. - Enrique Pérez Herrero, Aug 13 2011

References

• F. Smarandache, "Properties of the Numbers", University of Craiova Archives, 1975; Arizona State University Special Collections, Tempe, AZ

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..100
• G. S. Kazandzidis, On a Conjecture of Moessner and a General Problem, Bull. Soc. Math. Grèce, Nouvelle Série - vol. 2, fasc. 1-2, pp. 23-30, 1961.
• Index to divisibility sequences

Crossrefs

Cf. A000142, A003991, A271574.

Row n=3 of A225816.

Programs

Magma

[Factorial(n)^3: n in [0..15]]; // Vincenzo Librandi, Jan 13 2012

Maple

seq((n!)^3, n=0..14), # Karol A. Penson, Jul 28 2013

Mathematica

Table[(n!)^3, {n, 0, 20}] (* Stefan Steinerberger, Apr 14 2006 *)

PARI

a(n)=n!^3 \\ Charles R Greathouse IV, Jan 12 2012

Formula

a(n) = det(S(i+3, j), 1 <= i, j <= n), where S(n, k) are Stirling numbers of the second kind. - Mircea Merca, Apr 04 2013

From Karol A. Penson, Jul 28 2013: (Start)

G.f. of hypergeometric type: sum(a(n)*z^n/(n!)^3, n = 0..infinity) = 1/(1-z);

Integral representation as n-th moment of a positive function w(x) on a positive halfaxis (solution of the Stieltjes moment problem), in Maple notation:

a(n) = int(x^n*w(x), x = 0..infinity), n >= 0, where w(x) = MeijerG([[], []], [[0, 0, 0]], []], x), w(0) = infinity, limit(w(x), x = infinity) = 0.

w(x) is monotonically decreasing over (0, infinity). The Meijer G function above cannot be represented by any other known special function. This solution of the Stieltjes moment problem is not unique.

Asymptotics: a(n) -> (1/16)*sqrt(2)*Pi^(3/2)*(32*n^2 + 8*n + 1)*(n)^(-1/2+3*n)*exp(-3*n), for n -> infinity. (End)

D-finite with recurrence: a(n) -n^3*a(n-1)=0. - R. J. Mathar, Feb 16 2020

From Amiram Eldar, Nov 09 2020: (Start)

a(n) = A000142(n)^3.

Sum_{n>=0} 1/a(n) = A271574. (End)

a(n) = [x^n] Product_{k=1..n} (1 + k^3*x). - Vaclav Kotesovec, Feb 19 2022