|
|
REFERENCES
| K. P. Bogart and J. Q. Longyear, Counting 3 by n Latin rectangles, Proc. Amer. Math. Soc., 54 (1976), 463-467.
L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 183.
I. Gessel, Counting three-line Latin rectangles, Lect. Notes Math, 1234(1986), 106-111. [From Vladimir Shevelev (shevelev(AT)bgu.ac.il), Mar 25 2010]
Goulden and Jackson, Combin. Enum., Wiley, 1983 p. 284.
S. M. Jacob, The enumeration of the Latin rectangle of depth three..., Proc. London Math. Soc., 31 (1928), 329-336.
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.
S. M. Kerawala, The asymptotic number of three-deep Latin rectangles, Bull. Calcutta Math. Soc., 39 (1947), 71-72.
Koichi, Yamamoto, An asymptotic series for the number of three-line Latin rectangles, J. Math. Soc. Japan 1, (1950). 226-241.
W. Moser. A generalization of Riordan's formula for 3Xn Latin rectangles, Discrete Math., 40, 311-313 [From Vladimir Shevelev (shevelev(AT)bgu.ac.il), Mar 25 2010]
J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 210.
V. S. Shevelev, Reduced Latin rectangles and square matrices with indentical sums in the rows and columns [Russian], Diskret. Mat., 4 (1992), no. 1, 91-110.
V. S. Shevelev, A generalized Riordan formula for three-rowed Latin rectangles and its applications, DAN of the Ukraine, 2 (1991), 8-12 (in Russian) [From Vladimir Shevelev (shevelev(AT)bgu.ac.il), Mar 25 2010]
Vladimir Shevelev, SPECTRUM OF PERMANENTS VALUES AND ITS EXTREMAL MAGNITUDES ..., Arxiv preprint arXiv:1104.4051, 2011.
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).
D. S. Stones, The many formulae 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.
|
|
|
MAPLE
| for n from 1 to 250 do t0:=0; for j from 0 to n do for k from 0 to n-j do t0:=t0 + (2^j/j!)*k!*binomial(-3*(k+1), n-k-j); od: od: t0:=n!*t0; lprint(n, t0); od:
Maple code for A000186 based on Eq. (30) of Riordan, p. 205. Eq. (30a) on p. 206 is wrong. - N. J. A. Sloane, Jan 21 2010. Thanks to Neven Juric for correcting an error in the definition of fU, Mar 01 2010
# A000166
unprotect(D);
D := proc(n) option remember; if n<=1 then 1-n else (n-1)*(D(n-1)+D(n-2)); fi; end;
[seq(D(n), n=0..30)];
# A000179
U := proc(n) if n<=1 then 1-n else
add ((-1)^k*(2*n)*binomial(2*n-k, k)*(n-k)!/(2*n-k), k=0..n); fi; end;
[seq(U(n), n=0..30)];
# funny A000179
fU := proc(n) global U; if n<=-2 then U(-n) elif abs(n)=1 then -1 elif n=0 then 2 else U(n); fi; end;
[seq(fU(n), n=-10..10)];
# A000186
K:=proc(n) local k; global D, fU; (1/2)*add( binomial(n, k)*D(n-k)*D(k)*fU(n-2*k), k=0..n ); end;
[seq(K(n), n=0..30)];
|
