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 A273464 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. 10
 1, 1, 3, 1, 9, 24, 18, 1, 18, 126, 434, 762, 630, 187, 1, 30, 387, 2814, 12699, 36894, 69242, 81936, 57672, 21432, 3135, 1, 45, 915, 11127, 90270, 515970, 2139120, 6523428, 14683401, 24256853, 28975770, 24383838, 13860321, 4966929, 989970, 81462, 1, 63 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,3 LINKS R. J. Mathar, Table of n, a(n) for n = 1..575 J. A. De Loera, J. Rambau, F. Santos, Further topics, in: Triangulations, vol 25 of Algor. Computat. Math. (2010), 433-511. R. J. Mathar, Lozenge tilings of the equilateral triangle, arXiv:1909.06336 [math.CO], 2019. Francisco Santos, The Cayley trick and triangulations of products of simplices,  arXiv:math/0312069 [math.CO], 2004. Francisco Santos, The Cayley trick and triangulations of products of simplices, Cont. Math. 374 (2005), 151-177. Wikipedia, Lozenge. FORMULA T(n,2) = 3*(n-1)*(n-2)*(3*n^2+3*n-4)/8 . - R. J. Mathar, May 24 2016 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 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 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 From Petros Hadjicostas, Sep 13 2019: (Start) Conjecture for rightmost terms: A122722(n) =  n! * T(n, n*(n+1)/2) for n >= 1. 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. Sum_{0 <= s <= 2*k} (-1)^s * binomial(2*k, s) * T(n-s, k) = A011781(k) for n >= 2*k+1. (End) EXAMPLE Triangle T(n,k) (with rows n >= 1 and columns k >= 0) begins as follows:   1;   1,  3;   1,  9,  24,   18;   1, 18, 126,  434,   762,   630,   187;   1, 30, 387, 2814, 12699, 36894, 69242, 81936, 57672, 21432, 3135;   ... CROSSREFS Cf. A045943 (column k=1), A011555, A011556, A011781, A122722, A326367 (k=2), A326368 (k=3), A326369 (k=4), A000124 (row lengths). Sequence in context: A160568 A157403 A225118 * A105951 A038202 A128415 Adjacent sequences:  A273461 A273462 A273463 * A273465 A273466 A273467 KEYWORD tabf,nonn AUTHOR R. J. Mathar, May 23 2016 STATUS approved

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Last modified March 30 12:10 EDT 2020. Contains 333125 sequences. (Running on oeis4.)