A001623 -id:A001623 - cp's OEIS frontend

A130077 Largest x such that 2^x divides A001623(n), the number of reduced three-line Latin rectangles.

0, 2, 1, 3, 4, 6, 5, 6, 7, 9, 8, 11, 13, 14, 12, 16, 15, 17, 16, 18, 19, 21, 20, 21, 22, 24, 23, 27, 27, 30, 27, 29, 31, 33, 32, 34, 35, 37, 36, 37, 38, 40, 39, 42, 44, 45, 43, 50, 46, 48, 47, 49, 50, 52, 51, 52, 53, 55, 54, 59, 58, 62, 58, 60, 63, 65, 64, 66, 67, 69, 68, 69, 70

Offset: 3

Douglas Stones (dssto1(AT)student.monash.edu.au), May 06 2007

nonn

John Riordan, A recurrence relation for three-line Latin rectangles, Amer. Math. Monthly, 59 (1952), pp. 159-162.
D. S. Stones, The many formulas for the number of Latin rectangles, Electron. J. Combin 17 (2010), A1.
D. S. Stones and I. M. Wanless, Divisors of the number of Latin rectangles, J. Combin. Theory Ser. A 117 (2010), 204-215.

Cf. A001623, A007814, A130078, A130079.

a001623(n) = n*(n-3)!*sum(i=0, n, sum(j=0, n-i, (-1)^j*binomial(3*i+j+2, j)<<(n-i-j)/(n-i-j)!)*i!);
a(n) = valuation(a001623(n), 2); \\ Michel Marcus, Oct 02 2017

a(n) = A007814(A001623(n)). - Michel Marcus, Oct 02 2017

A130078 Largest 2^x dividing A001623(n), the number of reduced three-line Latin rectangles.

Original entry on oeis.org

1, 4, 2, 8, 16, 64, 32, 64, 128, 512, 256, 2048, 8192, 16384, 4096, 65536, 32768, 131072, 65536, 262144, 524288, 2097152, 1048576, 2097152, 4194304, 16777216, 8388608, 134217728, 134217728, 1073741824, 134217728, 536870912, 2147483648

Offset: 3

Douglas Stones (dssto1(AT)student.monash.edu.au), May 06 2007

nonn

John Riordan, A recurrence relation for three-line Latin rectangles, Amer. Math. Monthly, 59 (1952), pp. 159-162.
D. S. Stones, The many formulas for the number of Latin rectangles, Electron. J. Combin 17 (2010), A1.
D. S. Stones and I. M. Wanless, Divisors of the number of Latin rectangles, J. Combin. Theory Ser. A 117 (2010), 204-215.

Cf. A001623, A006519, A130077, A130079.

a001623(n) = n*(n-3)!*sum(i=0, n, sum(j=0, n-i, (-1)^j*binomial(3*i+j+2, j)<<(n-i-j)/(n-i-j)!)*i!);
a(n) = 2^valuation(a001623(n), 2); \\ Michel Marcus, Oct 02 2017

a(n) = A006519(A001623(n)). - Michel Marcus, Oct 02 2017

A130079 a(n) = n - A130077(n), i.e., n minus the largest x such that 2^x divides A001623(n), the number of reduced three-line Latin rectangles.

Original entry on oeis.org

3, 2, 4, 3, 3, 2, 4, 4, 4, 3, 5, 3, 2, 2, 5, 2, 4, 3, 5, 4, 4, 3, 5, 5, 5, 4, 6, 3, 4, 2, 6, 5, 4, 3, 5, 4, 4, 3, 5, 5, 5, 4, 6, 4, 3, 3, 6, 0, 5, 4, 6, 5, 5, 4, 6, 6, 6, 5, 7, 3, 5, 2, 7, 6, 4, 3, 5, 4, 4, 3, 5, 5, 5, 4, 6, 4, 1, 3, 6, 4, 5, 4, 6, 5, 5, 4, 6, 6, 6, 5, 7, 4, 5, 3, 7, 6, 5, 4

Offset: 3

Douglas Stones (dssto1(AT)student.monash.edu.au), May 06 2007

nonn

John Riordan, A recurrence relation for three-line Latin rectangles, Amer. Math. Monthly, 59 (1952), pp. 159-162.
D. S. Stones, The many formulas for the number of Latin rectangles, Electron. J. Combin 17 (2010), A1.
D. S. Stones and I. M. Wanless, Divisors of the number of Latin rectangles, J. Combin. Theory Ser. A 117 (2010), 204-215.

Cf. A001623, A130077, A130078.

a001623(n) = n*(n-3)!*sum(i=0, n, sum(j=0, n-i, (-1)^j*binomial(3*i+j+2, j)<<(n-i-j)/(n-i-j)!)*i!);
a(n) = n - valuation(a001623(n), 2); \\ Michel Marcus, Oct 02 2017

A001044 a(n) = (n!)^2.

Original entry on oeis.org

1, 1, 4, 36, 576, 14400, 518400, 25401600, 1625702400, 131681894400, 13168189440000, 1593350922240000, 229442532802560000, 38775788043632640000, 7600054456551997440000, 1710012252724199424000000, 437763136697395052544000000, 126513546505547170185216000000

Offset: 0

N. J. A. Sloane and R. K. Guy

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 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
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. - Wolfdieter Lang, Jan 09 2012
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
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
From Jerrold Grossman, Jul 22 2018: (Start)
a(n) is the number of n X n 0-1 matrices whose row sums and column sums are both {1,2,...,n}.
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.
(End)
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
Number of permutations P of length 2n which maximize Sum_{i=1..2n} |P_i - i|. - Fang Lixing, Dec 07 2018

			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_, Apr 06 2003
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
1 + x + 4*x^2 + 36*x^3 + 576*x^4 + 14400*x^5 + 518400*x^6 + ...

Archimedeans Problems Drive, Eureka, 22 (1959), 15.
David Burton, "The History of Mathematics", Sixth Edition, Problem 2, p. 433.
J. Dezert, editor, Smarandacheials, Mathematics Magazine, Aurora, Canada, No. 4/2004 (to appear).
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.
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).

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.
Daniel Dockery, Polygorials, Special "Factorials" of Polygonal Numbers, preprint, 2003.
R. K. Guy, Letters to N. J. A. Sloane, June-August 1968.
G. S. Kazandzidis, On a Conjecture of Moessner and a General Problem, Bull. Soc. Math. Grèce, Nouvelle Série - 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. [Annotated scanned copy]
T. Khovanova and W. Zhao, Mathematics of a Sudo-Kurve, arXiv:1808.06713 [math.HO], 2018.
S. Kitaev and J. Remmel, Classifying descents according to parity, Annals of Combinatorics, 11, 2007, 173-193.
Rob Pratt (Proposer), Problem 11573, Amer. Math. Monthly, 120 (2013), 372.
Luis Manuel Rivera, Integer sequences and k-commuting permutations, arXiv preprint arXiv:1406.3081 [math.CO], 2014-2015.
Simone Severini, Home page
Index to divisibility sequences
Index entries for sequences related to factorial numbers

Cf. A000142, A000292, A084939, A084940, A084941, A084942, A084943, A084944, A020549, A046032, A048617.
First right-hand column of triangle A008955.
Cf. A134434, A134435, A000442, A134375.
Row n=2 of A225816.
Cf. A000290.
With signs, a row of A288580.

List([0..20],n->Factorial(n)^2); # Muniru A Asiru, Oct 24 2018

import Data.List (genericIndex)
a001044 n = genericIndex a001044_list n
a001044_list = 1 : zipWith (*) (tail a000290_list) a001044_list
-- Reinhard Zumkeller, Sep 05 2015

[Factorial(n)^2: n in [0..20]]; // Vincenzo Librandi, Oct 24 2018

seq((n!)^2,n=0..20); # Dennis P. Walsh, Nov 26 2012

Table[n!^2, {n, 0, 20}] (* Stefan Steinerberger, Apr 07 2006 *)
Join[{1},Table[Det[DiagonalMatrix[Range[n]^2]],{n,20}]] (* Harvey P. Dale, Mar 31 2020 *)

a(n)=n!^2 \\ Charles R Greathouse IV, Jun 15 2011

import math
for n in range(0,20): print(math.factorial(n)**2, end=', ') # Stefano Spezia, Oct 29 2018

a(n) = Integral_{x>=0} 2*BesselK(0, 2*sqrt(x))*x^n. This integral represents the n-th moment of a positive function defined on the positive half-axis. - 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_{i=0..n-1} (2*i + 2) = 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)!. - Philippe Deléham, Jan 07 2004
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
D-finite with recurrence: a(0) = 1, a(n) = n^2*a(n-1). - Arkadiusz Wesolowski, Oct 04 2011
From Sergei N. Gladkovskii, Jun 14 2012: (Start)
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).
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.
(End).
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
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
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
a(n) = A000290(A000142(n)). - Michel Marcus, Nov 12 2013
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
From Ivan N. Ianakiev, Aug 16 2016: (Start)
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.
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.
(End).
From Ilya Gutkovskiy, Aug 16 2016: (Start)
a(n) = A184877(n)*A184877(n-1).
Sum_{n>=0} (-1)^n/a(n) = BesselJ(0,2) = A091681. (End)
Sum_{n>=0} a(n)/(2*n+1)! = 2*Pi/sqrt(27). - Daniel Suteu, Feb 06 2017
a(n) = [x^n] Product_{k=1..n} (1 + k^2*x). - Vaclav Kotesovec, Feb 19 2022
a(n) = (2*n+1)! * [x^(2*n+1)] 4*arcsin(x/2)/sqrt(4-x^2). - Ira M. Gessel, Dec 10 2024

More terms from James Sellers, Sep 19 2000

More terms from Simone Severini, Feb 15 2006

A000186 Number of 3 X n Latin rectangles in which the first row is in order.

Original entry on oeis.org

1, 0, 0, 2, 24, 552, 21280, 1073760, 70299264, 5792853248, 587159944704, 71822743499520, 10435273503677440, 1776780700509416448, 350461958856515690496, 79284041282622163140608, 20392765404792755583221760, 5917934230798104348783083520, 1924427226324694427836833857536

Offset: 0

N. J. A. Sloane

Or number of n X n matrices with exactly one 1 and one 2 in each row and column which are not in the main diagonal, other entries 0. - Vladimir Shevelev, Mar 22 2010

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 183.
Dulmage, A. L.; McMaster, G. E. A formula for counting three-line Latin rectangles. Proceedings of the Sixth Southeastern Conference on Combinatorics, Graph Theory and Computing (Florida Atlantic Univ., Boca Raton, Fla., 1975), pp. 279-289. Congressus Numerantium, No. XIV, Utilitas Math., Winnipeg, Man., 1975. MR0392611 (52 #13428). - From N. J. A. Sloane, Apr 06 2012
I. Gessel, Counting three-line Latin rectangles, Lect. Notes Math, 1234(1986), 106-111. [From Vladimir Shevelev, Mar 25 2010]
Goulden and Jackson, Combin. Enum., Wiley, 1983 p. 284.
S. M. Jacob, The enumeration of the Latin rectangle of depth three..., Proc. London Math. Soc., 31 (1928), 329-336.
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. M. Kerawala, The asymptotic number of three-deep Latin rectangles, Bull. Calcutta Math. Soc., 39 (1947), 71-72.
Koichi, Yamamoto, An asymptotic series for the number of three-line Latin rectangles, J. Math. Soc. Japan 1, (1950). 226-241.
W. Moser. A generalization of Riordan's formula for 3Xn Latin rectangles, Discrete Math., 40, 311-313 [From Vladimir Shevelev, Mar 25 2010]
J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 210.
V. S. Shevelev, Reduced Latin rectangles and square matrices with identical sums in the rows and columns [Russian], Diskret. Mat., 4 (1992), no. 1, 91-110.
V. S. Shevelev, A generalized Riordan formula for three-rowed Latin rectangles and its applications, DAN of the Ukraine, 2 (1991), 8-12 (in Russian) [From Vladimir Shevelev, Mar 25 2010]
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).
D. S. Stones, The many formulas for the number of Latin rectangles, Electron. J. Combin 17 (2010), A1.
RJ Stones, S Lin, X Liu, G Wang, On Computing the Number of Latin Rectangles, Graphs and Combinatorics, Graphs and Combinatorics (2016) 32:1187-1202; DOI 10.1007/s00373-015-1643-1

N. J. A. Sloane, Table of n, a(n) for n = 0..207
F. Alayont and N. Krzywonos, Rook Polynomials in Three and Higher Dimensions, 2012. - From _N. J. A. Sloane_, Jan 02 2013
K. P. Bogart and J. Q. Longyear, Counting 3 by n Latin rectangles, Proc. Amer. Math. Soc., 54 (1976), 463-467.
S. M. Jacob, The enumeration of the Latin rectangle of depth three..., Proc. London Math. Soc., 31 (1928), 329-336.
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. [Annotated scanned copy]
S. M. Kerawala, The asymptotic number of three-deep Latin rectangles, Bull. Calcutta Math. Soc., 39 (1947), 71-72. [Annotated scanned copy]
John Riordan, Table of a(0) - a(25), circa 1968
Vladimir Shevelev, Spectrum of permanent's values and its extremal magnitudes in Lambda_n^3 and Lambda_n(alpha, beta, gamma), arXiv preprint arXiv:1104.4051[math.CO], 2011.
D. S. Stones and I. M. Wanless, Divisors of the number of Latin rectangles, J. Combin. Theory Ser. A 117 (2010), 204-215.
Index entries for sequences related to Latin squares and rectangles

Cf. A000512, A000166, A000179, A335700, A170904.

for n from 1 to 250 do t0:=0; for j from 0 to n do for k from 0 to n-j do t0:=t0 + (2^j/j!)*k!*binomial(-3*(k+1), n-k-j); od: od: t0:=n!*t0; lprint(n,t0); od:
Maple code for A000186 based on Eq. (30) of Riordan, p. 205. Eq. (30a) on p. 206 is wrong. - N. J. A. Sloane, Jan 21 2010. Thanks to Neven Juric for correcting an error in the definition of fU, Mar 01 2010. Additional comment and modifications of code due to changes in underlying sequences from William P. Orrick, Aug 12 2020: Eq. (30) and Eq. (30a) are, in fact, related to each other by a trivial transformation and are both valid. Current code is based on Eq. (30a).
# A000166
unprotect(D);
D := proc(n) option remember; if n<=1 then 1-n else (n-1)*(D(n-1)+D(n-2));fi; end;
[seq(D(n), n=0..30)];
# A000179
U := proc(n) if n=0 then 1 else add ((-1)^k*(2*n)*binomial(2*n-k, k)*(n-k)!/(2*n-k), k=0..n); fi; end;
[seq(U(n), n=0..30)];
# A000186
K:=proc(n) local k; global D, U; add( binomial(n,k)*D(n-k)*D(k)*U(n-2*k), k=0..floor(n/2) ); end;
[seq(K(n), n=0..30)];
# another Maple program:
a:= proc(n) option remember; `if`(n<5, [1, 0, 0, 2, 24][n+1],
     ((n-1)*(n^2-2*n+2)*a(n-1) +(n-1)*(n-2)*(n^2-2*n+2)*a(n-2)
      +(n-1)*(n-2)*(n^2-2*n-2) *a(n-3)
      +2*(n-1)*(n-2)*(n-3)*(n^2-5*n+3) *a(n-4)
      -4*(n-2)*(n-3)*(n-4)*(n-1)^2 *a(n-5)) / (n-2))
    end:
seq(a(n), n=0..25);  # Alois P. Heinz, Nov 02 2013

a[n_] := (t0 = 0; Do[t0 = t0 + (2^j/j!)*k!*Binomial[-3*(k+1), n-k-j], {j, 0, n}, {k, 0, n-j}]; n!*t0); Table[a[n], {n, 0, 18}] (* Jean-François Alcover, Oct 13 2011, after Maple *)

# after Maple code based on Riordan's Eq. (30a)
d = [1,0]
for j in range(2,31):
    d.append((j - 1) * (d[-1] + d[-2]))
def u(n):
    if n == 0:
        return 1
    else:
        return sum((-1)^k * (2 * n) * binomial(2 * n - k, k) * factorial(n - k) / (2 * n - k) for k in range(0, n + 1))
def k(n):
    return sum(binomial(n, k) * d[n - k] * d[k] * u(n - 2 * k) for k in range(0, floor(n / 2) + 1))
[k(n) for n in range(0, 31)] # William P. Orrick, Aug 12 2020

a(n) = n!*Sum_{k+j<=n} (2^j/j!)*k!*binomial(-3*(k+1), n-k-j).
Note that the formula Sum_{k=0..n, k <= n/2} binomial(n, k)*D(n-k)*D(k)*U(n-2*k), where D() = A000166 and U() represents the menage numbers given by Riordan, p. 209 gives the wrong answers unless we set U(1) = -1 (or in other words we must take U() = A000179). With U(1) = 0 (see A335700) it produces A170904. See the Maple code here. - N. J. A. Sloane, Jan 21 2010, Apr 04 2010. Thanks to Vladimir Shevelev for clarifying this comment. Additional changes from William P. Orrick, Aug 12 2020
E.g.f.: exp(2*x) Sum(n>=0; n! x^n /(1+x)^(3*n+3)) from Gessel reference. - Wouter Meeussen, Nov 02 2013
a(n) ~ n!^2/exp(3). - Vaclav Kotesovec, Sep 08 2016
a(n+p)-2*a(n) is divisible by p for any prime p. - Mark van Hoeij, Jun 13 2019

Formula and more terms from Vladeta Jovovic, Mar 31 2001

Edited by N. J. A. Sloane, Jan 21 2010, Mar 04 2010, Apr 04 2010

A001009 Triangle giving number L(n,k) of normalized k X n Latin rectangles.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 3, 4, 4, 1, 11, 46, 56, 56, 1, 53, 1064, 6552, 9408, 9408, 1, 309, 35792, 1293216, 11270400, 16942080, 16942080, 1, 2119, 1673792, 420909504, 27206658048, 335390189568, 535281401856, 535281401856, 1, 16687, 103443808

Offset: 1

Brendan McKay

CRC Handbook of Combinatorial Designs, 1996, p. 104.

Herman Jamke, Table of n, a(n) for n = 1..66
Thomas Bloom, Problem 725, Erdős Problems.
Eric Fernando Bravo, On concatenations of Padovan and Perrin numbers, Math. Commun. (2023) Vol 28, 105-119.
Brendan D. McKay and Eric Rogoyski , Latin squares of order 10, Electron. J. Combinatorics, 2 (1995) #N3.
Douglas S. Stones, The many formulas for the number of Latin rectangles, Electron. J. Combin 17 (2010), A1.
Eric Weisstein's World of Mathematics, Latin Rectangle.
Index entries for sequences related to Latin squares and rectangles

Rows include A001623, A000573. Diagonals include A000576.

More terms from Herman Jamke (hermanjamke(AT)fastmail.fm), Sep 12 2010

A001626 Number of 3-line Latin rectangles.

Original entry on oeis.org

0, 0, 2, 36, 840, 29680, 1429920, 90318144, 7237943552, 717442928640, 86171602072320, 12331048749268480, 2072725870491859968, 404352831489304049664, 90605920564322676531200, 23110943021722435879157760, 6657484407493222296916131840

Offset: 1

N. J. A. Sloane

nonn

S. M. Jacob, The enumeration of the Latin rectangle of depth three..., Proc. London Math. Soc., 31 (1928), 329-336.
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.
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).

S. M. Jacob, The enumeration of the Latin rectangle of depth three..., Proc. London Math. Soc., 31 (1928), 329-336.
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. [Annotated scanned copy]
Index entries for sequences related to Latin squares and rectangles

Cf. A000186.

a(1) = 0, a(n) = A000186(n) + 2*(n-1)*a(n-1), n > 1. - Sean A. Irvine, Sep 25 2015

More terms from Sean A. Irvine, Sep 25 2015

A275921 Number of 5 X n Latin rectangles.

Original entry on oeis.org

56, 9408, 11270400, 27206658048, 112681643083776, 746988383076286464, 7533492323047902093312, 111048869433803210653040640, 2315236533572491933131807916032, 66415035616070432053233927044726784, 2560483881619577552584872021599994249216

Offset: 5

N. J. A. Sloane, Aug 28 2016

nonn

N. J. A. Sloane, Table of n, a(n) for n = 5..40 [Moused from the table in Stones et al. 2016]
R. J. Stones, S. Lin, X. Liu, G. Wang, On Computing the Number of Latin Rectangles, Graphs and Combinatorics (2016) 32:1187-1202; DOI 10.1007/s00373-015-1643-1

Cf. A000179, A000186, A001623.

A001624 Related to Latin rectangles.

Original entry on oeis.org

1, 5, 58, 1274, 41728, 1912112, 116346400, 9059742176, 877746364288, 103483282967936, 14581464284095744, 2419278174185319680, 466730664414683625472, 103580258158369503481856, 26198788829773597178540032

Offset: 2

N. J. A. Sloane

nonn

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.
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).

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. [Annotated scanned copy]
Index entries for sequences related to Latin squares and rectangles

a(2) = 1, a(n) = A001626(n) + A001626(n-1) + A001627(n-1) + (n-2)(a(n-1) + A001625(n-1)). - Sean A. Irvine, Sep 25 2015

More terms from Sean A. Irvine, Sep 25 2015

A001625 Related to Latin rectangles.

Original entry on oeis.org

2, 4, 60, 1276, 41888, 1916064, 116522048, 9069595840, 878460379392, 103547791177216, 14588580791234048, 2420219602973093376, 466877775127725240320, 103607067936116866084864, 26204424894484840874483712

Offset: 2

N. J. A. Sloane

nonn

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.
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).

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. [Annotated scanned copy]
Index entries for sequences related to Latin squares and rectangles

a(2) = 2, a(n) = A001626(n) + 2 * A001627(n-1) + 2 * (n-1) * A001624(n-1). - Sean A. Irvine, Sep 25 2015

More terms from Sean A. Irvine, Sep 25 2015

cp's OEIS Frontend

A130077 Largest x such that 2^x divides A001623(n), the number of reduced three-line Latin rectangles.

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A130078 Largest 2^x dividing A001623(n), the number of reduced three-line Latin rectangles.

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A130079 a(n) = n - A130077(n), i.e., n minus the largest x such that 2^x divides A001623(n), the number of reduced three-line Latin rectangles.

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A001044 a(n) = (n!)^2.

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A000186 Number of 3 X n Latin rectangles in which the first row is in order.

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A001009 Triangle giving number L(n,k) of normalized k X n Latin rectangles.

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A001626 Number of 3-line Latin rectangles.

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A275921 Number of 5 X n Latin rectangles.

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