cp's OEIS Frontend

A337303 Number of X-based filling of diagonals in a diagonal Latin square of order n.

%I A337303 #16 Apr 15 2023 00:24:00
%S A337303 1,1,0,0,96,480,57600,403200,191600640,1724405760,1597368729600,
%T A337303 17571056025600,28378507272192000,368920594538496000,
%U A337303 952903592436341145600,14293553886545117184000,55442575636536644075520000,942523785821122949283840000,5231730206388249282710863872000
%N A337303 Number of X-based filling of diagonals in a diagonal Latin square of order n.
%C A337303 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.
%H A337303 Andrew Howroyd, <a href="/A337303/b337303.txt">Table of n, a(n) for n = 0..100</a>
%H A337303 S. Kochemazov, O. Zaikin, E. Vatutin E., and A. Belyshev, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL23/Zaikin/zaikin3.html">Enumerating Diagonal Latin Squares of Order Up to 9</a>, Journal of Integer Sequences. Vol. 23. Iss. 1. 2020. Article 20.1.2.
%H A337303 E. I. Vatutin, <a href="https://vk.com/wall162891802_1291">About the number of X-based fillings of diagonals in a diagonal Latin squares of orders 1-15</a> (in Russian).
%H A337303 E. I. Vatutin, A. D. Belyshev, N. N. Nikitina, and M. O. Manzuk, <a href="http://evatutin.narod.ru/evatutin_dls_scf_gen.pdf">Use of X-based diagonal fillings and ESODLS CMS schemes for enumeration of main classes of diagonal Latin squares</a>, Telecommunications, 2023, No. 1, pp. 2-16, DOI: 10.31044/1684-2588-2023-0-1-2-16 (in Russian).
%H A337303 <a href="https://oeis.org/index/La#Latin">Index entries for sequences related to Latin squares and rectangles</a>.
%F A337303 a(n) = A337302(n)*n!.
%F A337303 a(n) = n!*A000316(floor(n/2)). - _Andrew Howroyd_, Mar 26 2023
%e A337303 One of the 96 X-based fillings of diagonals of a diagonal Latin square for order n=4:
%e A337303 1 . . 0
%e A337303 . 0 1 .
%e A337303 . 3 2 .
%e A337303 2 . . 3
%o A337303 (PARI) \\ here b(n) is A000459.
%o A337303 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)! )); }
%o A337303 a(n) = {2^(n\2) * b(n\2) * n!} \\ _Andrew Howroyd_, Mar 26 2023
%Y A337303 Cf. A000316, A000459, A309283, A274171, A337302.
%K A337303 nonn
%O A337303 0,5
%A A337303 _Eduard I. Vatutin_, Aug 22 2020
%E A337303 a(0)=1 prepended and terms a(16) and beyond from _Andrew Howroyd_, Mar 26 2023