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A001044
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(n!)^2.
(Formerly M3666 N1492)
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61
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1, 1, 4, 36, 576, 14400, 518400, 25401600, 1625702400, 131681894400, 13168189440000, 1593350922240000, 229442532802560000, 38775788043632640000, 7600054456551997440000, 1710012252724199424000000
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,3
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COMMENTS
| Let M_n be the symmetrical n X n matrix M_n(i,j)=1/Max(i,j); then for n>0 det(M_n)=1/a(n) - Benoit Cloitre, Apr 27 2002
The n-th entry of the sequence is the value of the permanent of a k by k matrix A defined as follows: k is the n-th odd number; if we concatenate the rows of A to form a vector v of length n^2, v_{i}=1 if i=1 or a multiple of 2. - Simone Severini, Feb 15 2006
a(n) = number of set partitions of {1,2,...,3n-1,3n} into blocks of size 3 in which the entries of each block mod 3 are distinct. For example, a(2) = 4 counts 123-456, 156-234, 126-345, 135-246. - David Callan, Mar 30 2007
From Emeric Deutsch, Nov 22 2007: (Start)
Number of permutations of {1,2,...,2n} with no even entry followed by a smaller entry. Example: a(2)=4 because we have 1234, 1324, 3124 and 2314.
Number of permutations of {1,2,...,2n} with n even entries that are followed by a smaller entry. Example: a(2)=4 because we have 2143, 3421, 4213 and 4321.
Number of permutations of {1,2,...,2n-1} with no even entry followed by a smaller entry. Example: a(2)=4 because we have 123,132,312 and 231.
Number of permutations of {1,2,...,2n-1} with n-1 odd entries followed by a smaller entry. Example: a(2)=4 because we have 132,312,231 and 321.
(End)
G. Leibniz in his "Ars Combinatoria" established the identity P(n)^2=P(n-1)[P(n+1)-P(n)], where P(n) = n!. (For example, see the Burton reference.) - Mohammad K. Azarian, Mar 28 2008
a(n) is also the determinant of the symmetric n X n matrix M defined by M(i,j) = sigma_2(gcd(i,j)) for 1 <= i,j <= n, and n>0, where sigma_2 is A001157 - Enrique Pérez Herrero, Aug 13 2011.
The o.g.f. of 1/a(n) is BesselI(0,2*sqrt(x)). See Abramowitz-Stegun (reference and link under A008277). p. 375, 9.6.10. [From Wolfdieter Lang, Jan 09 2012]
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REFERENCES
| Archimedeans Problems Drive, Eureka, 22 (1959), 15.
J. Dezert, editor, Smarandacheials, Mathematics Magazine, Aurora, Canada, No. 4/2004 (to appear).
G. S. Kazandzidis, On a Conjecture of Moessner and a General Problem, Bull. Soc. Math. Grece, Nouvelle Serie - vol. 2, fasc. 1-2, pp. 23-30.(1961)
S. M. Kerawala, The enumeration of the Latin rectangle of depth three by means of a difference equation, Bull. Calcutta Math. Soc., 33 (1941), 119-127.
S. Kitaev and J. Remmel, Classifying descents according to parity, Annals of Combinatorics, 11, 2007, 173-193.
J. Riordan, Combinatorial Identities, Wiley, 1968, p. 217.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
F. Smarandache, Back and Forth Factorials, Arizona State Univ., Special Collections, 1972.
R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Problem 5.62(b).
David Burton, "The History of Mathematics", Sixth Edition, Problem 2, p. 433.
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LINKS
| T. D. Noe, Table of n, a(n) for n = 0..100
P. J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.
J. Dezert, Smarandacheials
Simone Severini, Title?
Index to divisibility sequences
Index entries for sequences related to factorial numbers
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FORMULA
| Integral representation as n-th moment of a positive function on a positive half-axis, in Maple notation: a(n)=int(x^n*2*BesselK(0, 2*sqrt(x)), x=0..infinity), n=0, 1... - Karol A. Penson, Oct 09 2001
a(n) ~ 2*pi*n*e^(-2*n)*n^(2*n) - Joe Keane (jgk(AT)jgk.org), Jun 07 2002
a(n) = Polygorial(n, 4) = A000142(n)/A000079(n)*A000165(n) = n!/2^n*product(2*i+2, i=0..n-1) = n!*pochhammer(1, n) = n!^2 - Daniel Dockery (peritus(AT)gmail.com) Jun 13, 2003
a(n) = Sum{k>=0, (-1)^k*C(n, k)^2*k!*(2*n-k)! }. - DELEHAM Philippe, Jan 07 2004
a(n) = !n!_1 = !n! = Prod_{i=0, 1, 2, ... .}_{0<|n-i|<=n}(n-i) = n(n-1)(n-2)...(2)(1)(-1)(-2)...(-n+2)(-n+1)(-n) = [(-1)^n][(n!)^2]. - J. Dezert (Jean.Dezert(AT)onera.fr), Mar 21 2004
a(0) = 1, a(n) = n^2*a(n-1). [Arkadiusz Wesolowski, Oct 04 2011]
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EXAMPLE
| Consider the square array
1 2 3 4 5 6...
2 4 6 8 10 12...
3 6 9 12 15 18 ...
4 8 12 16 20 24...
5 10 15 20 25 30...
...
then a(n) = product of n-th antidiagonal. - Amarnath Murthy (amarnath_murthy(AT)yahoo.com), Apr 06 2003
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MATHEMATICA
| Table[n!^2, {n, 0, 20}] - Stefan Steinerberger, Apr 07 2006
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PROG
| (PARI) a(n)=n!^2 \\ Charles R Greathouse IV, Jun 15 2011
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CROSSREFS
| Cf. A000142, A000292, A084939, A084940, A084941, A084942, A084943, A084944, A020549, A046032, A048617.
First right-hand column of triangle A008955.
Cf. A134434, A134435, A000442, A134375.
Sequence in context: A132687 A073852 A139033 * A086879 A002761 A002084
Adjacent sequences: A001041 A001042 A001043 * A001045 A001046 A001047
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KEYWORD
| nonn,easy,nice
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com), R. K. Guy
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EXTENSIONS
| More terms from James A. Sellers (sellersj(AT)math.psu.edu), Sep 19 2000
More terms from Simone Severini (simoseve(AT)gmail.com), Feb 15 2006
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