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A001623 Number of 3 X n reduced (normalized) Latin rectangles.
(Formerly M3682 N1502)
12
1, 4, 46, 1064, 35792, 1673792, 103443808, 8154999232, 798030483328, 94866122760704, 13460459852344064, 2246551018310998016, 435626600453967929344, 97108406689489312301056, 24658059294992101453262848, 7075100096781964808223653888, 2277710095706779480096994066432, 817555425148510266964075644059648 (list; graph; refs; listen; history; text; internal format)
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.

Index entries for sequences related to Latin squares and rectangles

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

Cf. A001009.

Sequence in context: A191870 A099023 A000657 * A188634 A210855 A002077

Adjacent sequences:  A001620 A001621 A001622 * A001624 A001625 A001626

KEYWORD

nonn,nice

AUTHOR

N. J. A. Sloane

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

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Last modified December 14 00:21 EST 2017. Contains 295976 sequences.