A001623 Number of 3 X n reduced (normalized) Latin rectangles.
1, 4, 46, 1064, 35792, 1673792, 103443808, 8154999232, 798030483328, 94866122760704, 13460459852344064, 2246551018310998016, 435626600453967929344, 97108406689489312301056, 24658059294992101453262848, 7075100096781964808223653888, 2277710095706779480096994066432, 817555425148510266964075644059648
Offset: 3
Examples
G.f. = x^3 + 4*x^4 + 46*x^5 + 1064*x^6 + 35792*x^7 + 1673792*x^8 + ...
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
Crossrefs
Cf. A001009.
Programs
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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
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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 *) -
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
Formula
a(n) ~ (n-1)!^2/exp(3) ~ 2*Pi*n^(2*n-1)/exp(2*n+3). - Vaclav Kotesovec, Sep 08 2016
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
Comments