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A001623
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Number of 3 X n reduced (normalized) Latin rectangles.
(Formerly M3682 N1502)
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13
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1, 4, 46, 1064, 35792, 1673792, 103443808, 8154999232, 798030483328, 94866122760704, 13460459852344064, 2246551018310998016, 435626600453967929344, 97108406689489312301056, 24658059294992101453262848, 7075100096781964808223653888, 2277710095706779480096994066432, 817555425148510266964075644059648
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OFFSET
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3,2
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COMMENTS
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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}.
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REFERENCES
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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).
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LINKS
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FORMULA
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a(n) ~ (n-1)!^2/exp(3) ~ 2*Pi*n^(2*n-1)/exp(2*n+3). - Vaclav Kotesovec, Sep 08 2016
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EXAMPLE
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G.f. = x^3 + 4*x^4 + 46*x^5 + 1064*x^6 + 35792*x^7 + 1673792*x^8 + ...
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MAPLE
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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):
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MATHEMATICA
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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 *)
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PROG
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(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
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CROSSREFS
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KEYWORD
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nonn,nice
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AUTHOR
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EXTENSIONS
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Better description Jul 15 1995
Mathematica program, more terms, better definition, comment and Stones link from Wouter Meeussen, Oct 27 2013
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STATUS
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approved
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