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A337303 Number of X-based filling of diagonals in a diagonal Latin square of order n. 2
1, 1, 0, 0, 96, 480, 57600, 403200, 191600640, 1724405760, 1597368729600, 17571056025600, 28378507272192000, 368920594538496000, 952903592436341145600, 14293553886545117184000, 55442575636536644075520000, 942523785821122949283840000, 5231730206388249282710863872000 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,5
COMMENTS
Used for getting strong canonical forms (SCFs) of the diagonal Latin squares and for fast enumerating of the diagonal Latin squares based on equivalence classes.
LINKS
S. Kochemazov, O. Zaikin, E. Vatutin E., and A. Belyshev, Enumerating Diagonal Latin Squares of Order Up to 9, Journal of Integer Sequences. Vol. 23. Iss. 1. 2020. Article 20.1.2.
E. I. Vatutin, A. D. Belyshev, N. N. Nikitina, and M. O. Manzuk, Use of X-based diagonal fillings and ESODLS CMS schemes for enumeration of main classes of diagonal Latin squares, Telecommunications, 2023, No. 1, pp. 2-16, DOI: 10.31044/1684-2588-2023-0-1-2-16 (in Russian).
FORMULA
a(n) = A337302(n)*n!.
a(n) = n!*A000316(floor(n/2)). - Andrew Howroyd, Mar 26 2023
EXAMPLE
One of the 96 X-based fillings of diagonals of a diagonal Latin square for order n=4:
1 . . 0
. 0 1 .
. 3 2 .
2 . . 3
PROG
(PARI) \\ here b(n) is A000459.
b(n) = {sum(m=0, n, sum(k=0, n-m, (-1)^k * binomial(n, k) * binomial(n-k, m) * 2^(2*k+m-n) * (2*n-2*m-k)! )); }
a(n) = {2^(n\2) * b(n\2) * n!} \\ Andrew Howroyd, Mar 26 2023
CROSSREFS
Sequence in context: A233710 A234083 A234076 * A216372 A183684 A251167
KEYWORD
nonn
AUTHOR
Eduard I. Vatutin, Aug 22 2020
EXTENSIONS
a(0)=1 prepended and terms a(16) and beyond from Andrew Howroyd, Mar 26 2023
STATUS
approved

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Last modified March 28 15:38 EDT 2024. Contains 371254 sequences. (Running on oeis4.)