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%I #67 Jan 19 2023 11:09:53
%S 1,1,8,216,13824,1728000,373248000,128024064000,65548320768000,
%T 47784725839872000,47784725839872000000,63601470092869632000000,
%U 109903340320478724096000000,241457638684091756838912000000,662559760549147780765974528000000,2236139191853373760085164032000000000
%N a(n) = (n!)^3.
%C Permanent of upper right n X n corner of multiplication table (A003991). - _Marc LeBrun_, Dec 11 2003
%C 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
%C 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
%D F. Smarandache, "Properties of the Numbers", University of Craiova Archives, 1975; Arizona State University Special Collections, Tempe, AZ
%H Vincenzo Librandi, <a href="/A000442/b000442.txt">Table of n, a(n) for n = 0..100</a>
%H G. S. Kazandzidis, <a href="http://www.hms.gr/apothema/?s=sa&i=20">On a Conjecture of Moessner and a General Problem</a>, Bull. Soc. Math. Grèce, Nouvelle Série - vol. 2, fasc. 1-2, pp. 23-30, 1961.
%H <a href="/index/Di#divseq">Index to divisibility sequences</a>
%F 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
%F From _Karol A. Penson_, Jul 28 2013: (Start)
%F G.f. of hypergeometric type: sum(a(n)*z^n/(n!)^3, n = 0..infinity) = 1/(1-z);
%F 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:
%F 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.
%F 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.
%F 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)
%F D-finite with recurrence: a(n) -n^3*a(n-1)=0. - _R. J. Mathar_, Feb 16 2020
%F From _Amiram Eldar_, Nov 09 2020: (Start)
%F a(n) = A000142(n)^3.
%F Sum_{n>=0} 1/a(n) = A271574. (End)
%F a(n) = [x^n] Product_{k=1..n} (1 + k^3*x). - _Vaclav Kotesovec_, Feb 19 2022
%p seq((n!)^3, n=0..14), # _Karol A. Penson_, Jul 28 2013
%t Table[(n!)^3, {n, 0, 20}] (* _Stefan Steinerberger_, Apr 14 2006 *)
%o (PARI) a(n)=n!^3 \\ _Charles R Greathouse IV_, Jan 12 2012
%o (Magma) [Factorial(n)^3: n in [0..15]]; // _Vincenzo Librandi_, Jan 13 2012
%Y Cf. A000142, A003991, A271574.
%Y Row n=3 of A225816.
%K nonn,easy
%O 0,3
%A R. Muller
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