%I A001044 M3666 N1492
%S A001044 1,1,4,36,576,14400,518400,25401600,1625702400,131681894400,
%T A001044 13168189440000,1593350922240000,229442532802560000,
%U A001044 38775788043632640000,7600054456551997440000,1710012252724199424000000
%N A001044 (n!)^2.
%C A001044 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 (benoit7848c(AT)orange.fr), 
               Apr 27 2002
%C A001044 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 (ss54(AT)york.ac.uk), Feb 15 2006
%C A001044 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 
               (callan(AT)stat.wisc.edu), Mar 30 2007
%C A001044 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. - Emeric Deutsch (deutsch(AT)duke.poly.edu), Nov 22 2007
%C A001044 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. - Emeric Deutsch (deutsch(AT)duke.poly.edu), Nov 22 2007
%C A001044 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. - Emeric Deutsch (deutsch(AT)duke.poly.edu), Nov 22 2007
%C A001044 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. - Emeric Deutsch (deutsch(AT)duke.poly.edu), Nov 22 2007
%C A001044 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 (azarian(AT)evansville.edu), Mar 28 2008
%D A001044 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%D A001044 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 
               (includes this sequence).
%D A001044 Archimedeans Problems Drive, Eureka, 22 (1959), 15.
%D A001044 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)
%D A001044 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 A001044 J. Riordan, Combinatorial Identities, Wiley, 1968, p. 217.
%D A001044 R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see 
               Problem 5.62(b).
%D A001044 J. Dezert, editor, Smarandacheials, Mathematics Magazine, Aurora, Canada, 
               No. 4/2004 (to appear).
%D A001044 F. Smarandache, Back and Forth Factorials, Arizona State Univ., Special 
               Collections, 1972.
%D A001044 S. Kitaev and J. Remmel, Classifying descents according to parity, Annals 
               of Combinatorics, 11, 2007, 173-193.
%D A001044 David Burton, "The History of Mathematics", Sixth Edition, Problem 2, 
               p. 433.
%H A001044 T. D. Noe, <a href="b001044.txt">Table of n, a(n) for n=0..100</a>
%H A001044 P. J. Cameron, <a href="http://www.cs.uwaterloo.ca/journals/JIS/index.html">
               Sequences realized by oligomorphic permutation groups</a>, J. Integ. 
               Seqs. Vol. 3 (2000), #00.1.5.
%H A001044 J. Dezert, <a href="http://www.gallup.unm.edu/~smarandache/Smarandacheials.htm">
               Smarandacheials</a>
%H A001044 <a href="Sindx_Fa.html#factorial">Index entries for sequences related 
               to factorial numbers</a>
%H A001044 Simone Severini, <a href="http://www-users.york.ac.uk/~ss54">Title?</
               a>
%F A001044 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 (penson(AT)lptl.jussieu.fr), 
               Oct 09 2001
%F A001044 a(n) ~ 2*pi*n*e^(-2*n)*n^(2*n) - Joe Keane (jgk(AT)jgk.org), Jun 07 2002
%F A001044 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
%F A001044 a(n) = Sum{k>=0, (-1)^k*C(n, k)^2*k!*(2*n-k)! }. - DELEHAM Philippe (kolotoko(AT)wanadoo.fr), 
               Jan 07 2004
%F A001044 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
%e A001044 Consider the square array
%e A001044 1 2 3 4 5 6...
%e A001044 2 4 6 8 10 12...
%e A001044 3 6 9 12 15 18 ...
%e A001044 4 8 12 16 20 24...
%e A001044 5 10 15 20 25 30...
%e A001044 ...
%e A001044 then a(n) = product of n-th antidiagonal. - Amarnath Murthy (amarnath_murthy(AT)yahoo.com), 
               Apr 06 2003
%p A001044 seq(add(count(Permutation(k))*count(Permutation(k+1)),k=0..n),n=0..14); 
               - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Oct 17 2006
%p A001044 a:=n->(mul( k^2, k=1..n)): seq(a(n), n=0..15); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), 
               Jan 26 2008
%p A001044 with (combstruct):a:=proc(m) [ZL, {ZL=Set(Cycle(Z, card>=m))}, labeled]; 
               end: ZLL:=a(1):seq(count(ZLL, size=n)*n!, n=0..15); - Zerinvary Lajos 
               (zerinvarylajos(AT)yahoo.com), Jun 11 2008
%t A001044 Table[n!^2, {n, 0, 20}] - Stefan Steinerberger (stefan.steinerberger(AT)gmail.com), 
               Apr 07 2006
%o A001044 (Other) SAGE:[stirling_number1(n,1)^2for n in xrange(1,17)] [From Zerinvary 
               Lajos (zerinvarylajos(AT)yahoo.com), Mar 14 2009]
%Y A001044 Cf. A000142, A000292, A084939, A084940, A084941, A084942, A084943, A084944.
%Y A001044 Cf. A020549, A046032, A048617.
%Y A001044 First right-hand column of triangle A008955.
%Y A001044 Cf. A134434, A134435.
%Y A001044 Sequence in context: A132687 A073852 A139033 this_sequence A086879 A002761 
               A002084
%Y A001044 Adjacent sequences: A001041 A001042 A001043 this_sequence A001045 A001046 
               A001047
%K A001044 nonn,easy,nice
%O A001044 0,3
%A A001044 N. J. A. Sloane (njas(AT)research.att.com), R. K. Guy
%E A001044 More terms from James A. Sellers (sellersj(AT)math.psu.edu), Sep 19 2000
%E A001044 More terms from Simone Severini (ss54(AT)york.ac.uk), Feb 15 2006