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%I M3666 N1492 #221 Apr 14 2024 17:51:43
%S 1,1,4,36,576,14400,518400,25401600,1625702400,131681894400,
%T 13168189440000,1593350922240000,229442532802560000,
%U 38775788043632640000,7600054456551997440000,1710012252724199424000000,437763136697395052544000000,126513546505547170185216000000
%N a(n) = (n!)^2.
%C 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
%C The n-th entry of the sequence is the value of the permanent of a k X 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
%C 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
%C From _Emeric Deutsch_, Nov 22 2007: (Start)
%C 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.
%C 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.
%C 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.
%C 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.
%C (End)
%C 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
%C 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
%C 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. - _Wolfdieter Lang_, Jan 09 2012
%C Number of n x n x n cubes C of zeros and ones such that C(x,y,z) and C(u,v,w) can be nonzero simultaneously only if either x!=u, y!=v, or z!=w. This generalizes permutations which can be considered as n x n squares P of zeros and ones such that P(x,y) and P(u,v) can be nonzero simultaneously only if either x!=u or y!=v. - _Joerg Arndt_, May 28 2012
%C a(n) is the number of functions f:[n]->[n(n+1)/2] such that, if round(sqrt(2f(x))) = round(sqrt(2f(y))), then x=y. - _Dennis P. Walsh_, Nov 26 2012
%C From _Jerrold Grossman_, Jul 22 2018: (Start)
%C a(n) is the number of n X n 0-1 matrices whose row sums and column sums are both {1,2,...,n}.
%C a(n) is the number of linear arrangements of 2n blocks of n different colors, 2 of each color, such that there are an even number of blocks between each pair of blocks of the same color.
%C (End)
%C Number of ways to place n instances of a digit inside an n X n X n cube so that no two instances lie on a plane parallel to a face of the cube (see Khovanova link, Lemma 6, p. 22). - _Tanya Khovanova_ and _Wayne Zhao_, Oct 17 2018
%C Number of permutations P of length 2n which maximize Sum_{i=1..2n} |P_i - i|. - _Fang Lixing_, Dec 07 2018
%D Archimedeans Problems Drive, Eureka, 22 (1959), 15.
%D David Burton, "The History of Mathematics", Sixth Edition, Problem 2, p. 433.
%D J. Dezert, editor, Smarandacheials, Mathematics Magazine, Aurora, Canada, No. 4/2004 (to appear).
%D 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.
%D J. Riordan, Combinatorial Identities, Wiley, 1968, p. 217.
%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%D F. Smarandache, Back and Forth Factorials, Arizona State Univ., Special Collections, 1972.
%D R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Problem 5.62(b).
%H T. D. Noe, <a href="/A001044/b001044.txt">Table of n, a(n) for n = 0..100</a>
%H P. J. Cameron, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL3/groups.html">Sequences realized by oligomorphic permutation groups</a>, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.
%H Daniel Dockery, <a href="https://web.archive.org/web/20140617132401/http://danieldockery.com/res/math/polygorials.pdf">Polygorials, Special "Factorials" of Polygonal Numbers</a>, preprint, 2003.
%H R. K. Guy, <a href="/A002186/a002186.pdf">Letters to N. J. A. Sloane, June-August 1968</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 S. M. Kerawala, <a href="/A001623/a001623.pdf">The enumeration of the Latin rectangle of depth three by means of a difference equation</a>, Bull. Calcutta Math. Soc., 33 (1941), 119-127. [Annotated scanned copy]
%H T. Khovanova and W. Zhao, <a href="https://arxiv.org/abs/1808.06713">Mathematics of a Sudo-Kurve</a>, arXiv:1808.06713 [math.HO], 2018.
%H S. Kitaev and J. Remmel, <a href="http://dx.doi.org/10.1007/s00026-007-0313-2">Classifying descents according to parity</a>, Annals of Combinatorics, 11, 2007, 173-193.
%H Rob Pratt (Proposer), <a href="http://www.jstor.org/stable/10.4169/amer.math.monthly.120.04.365">Problem 11573</a>, Amer. Math. Monthly, 120 (2013), 372.
%H Luis Manuel Rivera, <a href="http://arxiv.org/abs/1406.3081">Integer sequences and k-commuting permutations</a>, arXiv preprint arXiv:1406.3081 [math.CO], 2014-2015.
%H Simone Severini, <a href="http://www-users.york.ac.uk/~ss54">Title?</a> [dead link]
%H <a href="/index/Di#divseq">Index to divisibility sequences</a>
%H <a href="/index/Fa#factorial">Index entries for sequences related to factorial numbers</a>
%F 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
%F a(n) ~ 2*Pi*n*e^(-2*n)*n^(2*n). - Joe Keane (jgk(AT)jgk.org), Jun 07 2002
%F a(n) = polygorial(n, 4) = A000142(n)/A000079(n)*A000165(n) = (n!/2^n)*Product_{i=0..n-1} (2*i + 2) = n!*Pochhammer(1, n) = n!^2. - Daniel Dockery (peritus(AT)gmail.com), Jun 13 2003
%F a(n) = Sum_{k>=0} (-1)^k*C(n, k)^2*k!*(2*n-k)!. - _Philippe Deléham_, Jan 07 2004
%F a(n) = !n!_1 = !n! = Product_{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
%F D-finite with recurrence: a(0) = 1, a(n) = n^2*a(n-1). - _Arkadiusz Wesolowski_, Oct 04 2011
%F From _Sergei N. Gladkovskii_, Jun 14 2012: (Start)
%F A(x) = Sum_{n>=0,N) a(n)*x^n = 1 + x/(U(0;N-2)-x); N >= 4; U(k)= 1 + x*(k+1)^2 - x*(k+2)^2/G(k+1); besides U(0;infinity)=x; (continued fraction, Euler's 1st kind, 1-step).
%F Let B(x) = Sum_{n>=0} a(n)*x^n/((n!)*(n+s)!), then B(0) = 1/(1-x) for abs(x) < 1 and B(1)= -1/x * log(1-x) for abs(x)< 1.
%F (End).
%F G.f.: 1 + x*(G(0) - 1)/(x-1) where G(k) = 1 - (k+1)^2*(1 - x*G(k+1)). - _Sergei N. Gladkovskii_, Jan 15 2013
%F a(n) = det(S(i+2,j), 1 <= i,j <= n), where S(n,k) are Stirling numbers of the second kind. - _Mircea Merca_, Apr 04 2013
%F a(n) = (2*n+1)!*2^(-4*n)*Sum_{k=0..n} (-1)^k*C(2*n+1,n-k)/(2*k+1). - _Mircea Merca_, Nov 12 2013
%F a(n) = A000290(A000142(n)). - _Michel Marcus_, Nov 12 2013
%F Sum_{n>=0} 1/a(n) = A070910 [Gradsteyn, Rzyhik 0.246.1]. - _R. J. Mathar_, Feb 25 2014. Corrected by _Ilya Gutkovskiy_, Aug 16 2016
%F From _Ivan N. Ianakiev_, Aug 16 2016: (Start)
%F a(n) = a(n-1) + 2*((n-1)^2)*sqrt(a(n-1)*a(n-2)) + ((n-1)^4)*a(n-2), for n > 1.
%F a(n) = a(n-1) - 2*(n^2 - 1)*sqrt(a(n-1)*a(n-2)) + (n^2 - 1)*a(n-2), for n > 1.
%F (End).
%F From _Ilya Gutkovskiy_, Aug 16 2016: (Start)
%F a(n) = A184877(n)*A184877(n-1).
%F Sum_{n>=0} (-1)^n/a(n) = BesselJ(0,2) = A091681. (End)
%F Sum_{n>=0} a(n)/(2*n+1)! = 2*Pi/sqrt(27). - _Daniel Suteu_, Feb 06 2017
%F a(n) = [x^n] Product_{k=1..n} (1 + k^2*x). - _Vaclav Kotesovec_, Feb 19 2022
%e Consider the square array
%e 1, 2, 3, 4, 5, 6, ...
%e 2, 4, 6, 8, 10, 12, ...
%e 3, 6, 9, 12, 15, 18, ...
%e 4, 8, 12, 16, 20, 24, ...
%e 5, 10, 15, 20, 25, 30, ...
%e ...
%e then a(n) = product of n-th antidiagonal. - _Amarnath Murthy_, Apr 06 2003
%e a(3) = 36 since there are 36 functions f:[3]->[6] such that, if round(sqrt(2f(x))) = round(sqrt(2f(y))), then x=y. The functions, denoted by <f(1),f(2),f(3)>, are <1,2,4>, <1,2,5>, <1,2,6>, <1,3,4>, <1,3,5>, <1,3,6> and their respective permutations. - _Dennis P. Walsh_, Nov 26 2012
%e 1 + x + 4*x^2 + 36*x^3 + 576*x^4 + 14400*x^5 + 518400*x^6 + ...
%p seq((n!)^2,n=0..20); # _Dennis P. Walsh_, Nov 26 2012
%t Table[n!^2, {n, 0, 20}] (* _Stefan Steinerberger_, Apr 07 2006 *)
%t Join[{1},Table[Det[DiagonalMatrix[Range[n]^2]],{n,20}]] (* _Harvey P. Dale_, Mar 31 2020 *)
%o (PARI) a(n)=n!^2 \\ _Charles R Greathouse IV_, Jun 15 2011
%o (Haskell)
%o import Data.List (genericIndex)
%o a001044 n = genericIndex a001044_list n
%o a001044_list = 1 : zipWith (*) (tail a000290_list) a001044_list
%o -- _Reinhard Zumkeller_, Sep 05 2015
%o (Magma) [Factorial(n)^2: n in [0..20]]; // _Vincenzo Librandi_, Oct 24 2018
%o (GAP) List([0..20],n->Factorial(n)^2); # _Muniru A Asiru_, Oct 24 2018
%o (Python) import math
%o for n in range(0,20): print(math.factorial(n)**2, end=', ') # _Stefano Spezia_, Oct 29 2018
%Y Cf. A000142, A000292, A084939, A084940, A084941, A084942, A084943, A084944, A020549, A046032, A048617.
%Y First right-hand column of triangle A008955.
%Y Cf. A134434, A134435, A000442, A134375.
%Y Row n=2 of A225816.
%Y Cf. A000290.
%Y With signs, a row of A288580.
%K nonn,easy,nice,changed
%O 0,3
%A _N. J. A. Sloane_, _R. K. Guy_
%E More terms from _James A. Sellers_, Sep 19 2000
%E More terms from _Simone Severini_, Feb 15 2006
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