0,3

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

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

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

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

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

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

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

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

Archimedeans Problems Drive, Eureka, 22 (1959), 15.

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.

J. Riordan, Combinatorial Identities, Wiley, 1968, p. 217.

R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Problem 5.62(b).

J. Dezert, editor, Smarandacheials, Mathematics Magazine, Aurora, Canada, No. 4/2004 (to appear).

F. Smarandache, Back and Forth Factorials, Arizona State Univ., Special Collections, 1972.

S. Kitaev and J. Remmel, Classifying descents according to parity, Annals of Combinatorics, 11, 2007, 173-193.

David Burton, "The History of Mathematics", Sixth Edition, Problem 2, p. 433.

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

Index entries for sequences related to factorial numbers

Simone Severini, Title?

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

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 (kolotoko(AT)wanadoo.fr), 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

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

seq(add(count(Permutation(k))*count(Permutation(k+1)), k=0..n), n=0..14); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Oct 17 2006

a:=n->(mul( k^2, k=1..n)): seq(a(n), n=0..15); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jan 26 2008

Table[n!^2, {n, 0, 20}] - Stefan Steinerberger (stefan.steinerberger(AT)gmail.com), Apr 07 2006

Cf. A000142, A000292, A084939, A084940, A084941, A084942, A084943, A084944.

Cf. A020549, A046032, A048617.

First right-hand column of triangle A008955.

Cf. A134434, A134435.

Adjacent sequences: A001041 A001042 A001043 this_sequence A001045 A001046 A001047

Sequence in context: A132687 A073852 A139033 this_sequence A086879 A002761 A002084

nonn,easy,nice

njas, R. K. Guy

More terms from James A. Sellers (sellersj(AT)math.psu.edu), Sep 19 2000

More terms from Simone Severini (ss54(AT)york.ac.uk), Feb 15 2006