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A347927 a(n) is the number of reduced Latin trapezoids of height 3, whose top row has n boxes, the middle row has n+1 boxes, and the bottom row has n+2 boxes.

%I A347927 #14 Oct 23 2021 17:37:07
%S A347927 1,6,68,1670,67295,3825722,285667270,26889145828,3102187523467,
%T A347927 429700007845870,70303573947346474,13405343287124139802,
%U A347927 2945521072579394529097,738633749151050116349946,209620243382776121032416188,66830750007674204750148252472,23780886787936166425634118631117
%N A347927 a(n) is the number of reduced Latin trapezoids of height 3, whose top row has n boxes, the middle row has n+1 boxes, and the bottom row has n+2 boxes.
%H A347927 Peter Luschny, <a href="/A347927/b347927.txt">Table of n, a(n) for n = 1..100</a>. Data from George Spahn and Doron Zeilberger, see link.
%H A347927 George Spahn and Doron Zeilberger, <a href="http://ecajournal.haifa.ac.il/Volume2022/ECA2022_S2A8.pdf">Automatic Counting of Generalized Latin Rectangles and Trapezoids</a>, Enumerative Combinatorics and Applications, 2:1 (2022).
%H A347927 George Spahn and Doron Zeilberger, <a href="https://sites.math.rutgers.edu/~zeilberg/tokhniot/oLatinTrapezoids1.txt">Latin trapezoids with three rows</a>, the first 100 terms.
%H A347927 George Spahn and Doron Zeilberger, <a href="https://sites.math.rutgers.edu/~zeilberg/tokhniot/LatinTrapezoids.txt">Latin trapezoids</a>, a Maple package.
%e A347927 There are 6 reduced Latin trapezoids of height 3 with base of length 4:
%e A347927 ----------------------------------------------
%e A347927     2, 3;       |    4, 3;       |    2, 3;
%e A347927    3, 1, 2;     |   3, 1, 2;     |   3, 4, 1;
%e A347927   1, 2, 3, 4;   |  1, 2, 3, 4;   |  1, 2, 3, 4;
%e A347927 -----------------------------------------------
%e A347927     2, 1;       |    2, 3;       |    2, 3;
%e A347927    3, 4, 2;     |   3, 4, 2;     |   4, 1, 2;
%e A347927   1, 2, 3, 4;   |  1, 2, 3, 4;   |  1, 2, 3, 4;
%e A347927 -----------------------------------------------
%Y A347927 Cf. A002860 (Latin squares), A000186, A001623, A001626.
%K A347927 nonn
%O A347927 1,2
%A A347927 _Peter Luschny_, Oct 22 2021