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%I M3682 N1502 #51 Nov 09 2016 15:03:10
%S 1,4,46,1064,35792,1673792,103443808,8154999232,798030483328,
%T 94866122760704,13460459852344064,2246551018310998016,
%U 435626600453967929344,97108406689489312301056,24658059294992101453262848,7075100096781964808223653888,2277710095706779480096994066432,817555425148510266964075644059648
%N Number of 3 X n reduced (normalized) Latin rectangles.
%C 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}.
%D 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 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H Robert Israel, <a href="/A001623/b001623.txt">Table of n, a(n) for n = 3..254</a>
%H S. M. Kerawala, <a href="/A001623/a001623.pdf">The enumeration of the Latin rectangle of depth three by means of a difference equation</a>, Bull. Calcutta Math. Soc., 33 (1941), 119-127. [Annotated scanned copy]
%H D. S. Stones, <a href="http://www.combinatorics.org/ojs/index.php/eljc/article/view/v17i1a1">The many formulas for the number of Latin rectangles</a>, Electron. J. Combin. 17 (2010), A1.
%H D. S. Stones and I. M. Wanless, <a href="http://dx.doi.org/10.1016/j.jcta.2009.03.019">Divisors of the number of Latin rectangles</a>, J. Combin. Theory Ser. A 117 (2010), 204-215.
%H R. J. Stones, S. Lin, X. Liu, G. Wang, <a href="http://dx.doi.org/10.1007/s00373-015-1643-1">On Computing the Number of Latin Rectangles</a>, Graphs and Combinatorics (2016) 32:1187-1202; DOI 10.1007/s00373-015-1643-1.
%H <a href="/index/La#Latin">Index entries for sequences related to Latin squares and rectangles</a>
%F a(n) ~ (n-1)!^2/exp(3) ~ 2*Pi*n^(2*n-1)/exp(2*n+3). - _Vaclav Kotesovec_, Sep 08 2016
%e G.f. = x^3 + 4*x^4 + 46*x^5 + 1064*x^6 + 35792*x^7 + 1673792*x^8 + ...
%p 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):
%p map(f, [$3..30]); # _Robert Israel_, Nov 07 2016
%t 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 *)
%o (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
%Y Cf. A001009.
%K nonn,nice
%O 3,2
%A _N. J. A. Sloane_
%E Better description Jul 15 1995
%E Mathematica program, more terms, better definition, comment and Stones link from _Wouter Meeussen_, Oct 27 2013
%E Minor corrections by _M. F. Hasler_, Oct 27 2013
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