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%I #79 Dec 12 2023 18:38:43
%S 1,1,3,1,9,24,18,1,18,126,434,762,630,187,1,30,387,2814,12699,36894,
%T 69242,81936,57672,21432,3135,1,45,915,11127,90270,515970,2139120,
%U 6523428,14683401,24256853,28975770,24383838,13860321,4966929,989970,81462,1,63
%N The number of tilings of an equilateral triangle of side length n with k lozenges and n^2 - 2*k unit triangles. Triangle T(n, k) with n >= 1 and 0 <= k <= n*(n + 1)/2, read by rows.
%H R. J. Mathar, <a href="/A273464/b273464.txt">Table of n, a(n) for n = 1..575</a>
%H J. A. De Loera, J. Rambau, F. Santos, <a href="http://dx.doi.org/10.1007/978-3-642-12971-1_9">Further topics</a>, in: Triangulations, vol 25 of Algor. Computat. Math. (2010), 433-511.
%H R. J. Mathar, <a href="https://arxiv.org/abs/1909.06336">Lozenge tilings of the equilateral triangle</a>, arXiv:1909.06336 [math.CO], 2019.
%H Francisco Santos, <a href="https://arxiv.org/abs/math/0312069">The Cayley trick and triangulations of products of simplices</a>, arXiv:math/0312069 [math.CO], 2004.
%H Francisco Santos, <a href="http://dx.doi.org/10.1090/conm/374/06904">The Cayley trick and triangulations of products of simplices</a>, Cont. Math. 374 (2005), 151-177.
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Lozenge">Lozenge</a>.
%F T(n,2) = 3*(n-1)*(n-2)*(3*n^2+3*n-4)/8 . - _R. J. Mathar_, May 24 2016
%F T(n,3) = (n-2)*(9*n^5-9*n^4-81*n^3+81*n^2+160*n-192)/16. - _Greg Dresden_, Jul 03 2019
%F Conjecture: T(n,4) = 3*(n-2)*(n-3)*(9*n^6+9*n^5-135*n^4-81*n^3+670*n^2+104*n-1216)/128. - _Greg Dresden_, Jul 03 2019
%F Conjecture: T(n,5) = 3*(n-3)*(n+3)* (27*n^8 -135*n^7 -387*n^6 +2835*n^5 -168*n^4 -18732*n^3 +19568*n^2 +36992*n -56320)/1280. - _R. J. Mathar_, Jul 07 2019
%F From _Petros Hadjicostas_, Sep 13 2019: (Start)
%F Conjecture for rightmost terms: A122722(n) = n! * T(n, n*(n+1)/2) for n >= 1.
%F Conjectures for column k >= 0: Sum_{0 <= s <= 2*k + 1} (-1)^s * binomial(2*k+1, s) * T(n-s, k) = 0 for n >= 2*k+2.
%F Sum_{0 <= s <= 2*k} (-1)^s * binomial(2*k, s) * T(n-s, k) = A011781(k) for n >= 2*k+1. (End)
%e Triangle T(n,k) (with rows n >= 1 and columns k >= 0) begins as follows:
%e 1;
%e 1, 3;
%e 1, 9, 24, 18;
%e 1, 18, 126, 434, 762, 630, 187;
%e 1, 30, 387, 2814, 12699, 36894, 69242, 81936, 57672, 21432, 3135;
%e ...
%Y Cf. A045943 (column k=1), A011555, A011556, A011781, A122722, A326367 (k=2), A326368 (k=3), A326369 (k=4), A000124 (row lengths).
%K tabf,nonn
%O 1,3
%A _R. J. Mathar_, May 23 2016
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