OFFSET
3,2
COMMENTS
A Latin rectangle [L_{n,k}] is called normalized [N_{n,k}] if the first row is (0,1, . . . , n-1), and reduced [R_{n,k}] if the first row is (0,1, . . . , n-1) and the first column is (0,1, . . . , k-1). Then L_{n,k} = n! N_{n,k} = (n! (n-1)! /(n-k)!) R_{n,k}.
REFERENCES
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).
LINKS
Robert Israel, Table of n, a(n) for n = 3..254
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]
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.
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.
FORMULA
a(n) ~ (n-1)!^2/exp(3) ~ 2*Pi*n^(2*n-1)/exp(2*n+3). - Vaclav Kotesovec, Sep 08 2016
EXAMPLE
G.f. = x^3 + 4*x^4 + 46*x^5 + 1064*x^6 + 35792*x^7 + 1673792*x^8 + ...
MAPLE
f:= n-> add(n*factorial(n-3)*factorial(i)*simplify(hypergeom([3*i+3, -n+i], [], 1/2))/(2^(-n+i)*factorial(n-i)), i=0..n):
map(f, [$3..30]); # Robert Israel, Nov 07 2016
MATHEMATICA
Table[Sum[ n (n - 3)! (-1)^j 2^(n -i-j) i!/(n-i-j)! Binomial[3 i + j + 2, j], {i, 0, n}, {j, 0, n - i} ], {n, 3, 25}] (* Wouter Meeussen, Oct 27 2013 *)
PROG
(PARI) 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!) \\ - M. F. Hasler, Oct 27 2013
CROSSREFS
KEYWORD
nonn,nice
AUTHOR
EXTENSIONS
Better description Jul 15 1995
Mathematica program, more terms, better definition, comment and Stones link from Wouter Meeussen, Oct 27 2013
Minor corrections by M. F. Hasler, Oct 27 2013
STATUS
approved