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%I #68 Aug 12 2022 19:23:06
%S 1,1,2,1,4,4,1,7,11,8,1,12,24,28,16,1,20,52,70,68,32,1,33,110,168,193,
%T 160,64,1,54,228,401,497,510,368,128,1,88,467,944,1257,1412,1304,832,
%U 256,1,143,949,2187,3172,3736,3879,3248,1856,512
%N Triangle read by rows: T(n,k) gives the number of domino towers of height k consisting of n bricks.
%C The towers must have a contiguous base of bricks, and each brick must be at least half supported below by another brick. The stacks do not need to be stable.
%C Conjecture: For n > 1, T(n,2) = A000071(n+2).
%C A038622(n-1,k) appears to give the number of domino towers consisting of n bricks with a base of k bricks.
%C Conjecture: T(n,n-1) = A339252(n-2). - _Peter Kagey_, Nov 21 2020
%C Conjecture: T(n,n-2) = A339254(n-3). - _Peter Luschny_, Nov 29 2020
%C Conjecture: T(n,n-3) = A339029(n-4). - _Peter Luschny_, Dec 01 2020
%C From _Peter Luschny_, Dec 01 2020: (Start)
%C The above conjectures can be summarized as follows:
%C T(2*n + k, n + k) = d_{n}(n + k - 1) for k >= 1 and 0 <= n <= 3, where
%C d_{0}(m) = 2^(m-1)*2;
%C d_{1}(m) = 2^(m-3)*(10 + 6*m);
%C d_{2}(m) = 2^(m-5)*(70 + 43*m + 9*m^2);
%C d_{3}(m) = 2^(m-7)*(588 + 367*m + 84*m^2 + 9*m^3). (End)
%D Miklos Bona, editor, Handbook of Enumerative Combinatorics, CRC Press, 2015, pages 25-27.
%H Peter Luschny, <a href="/A333650/b333650.txt">Table of n, a(n), for row(k) for k = 1..18</a> (the first 14 rows by Peter Kagey).
%H J. Bétréma and J.-G. Penaud, <a href="https://doi.org/10.1016/0304-3975(93)90304-C">Animaux et arbres guingois</a>, Theoretical Computer Science 117, 67-89, 1993.
%H D. Gouyou-Beauchamps and G. Viennot, <a href="https://doi.org/10.1016/0196-8858(88)90017-6">Equivalence of the two dimensional directed animals problem to a one-dimensional path problem</a>, Adv. in Appl. Math. 9(3), 334-357, 1988.
%H Peter Kagey, <a href="https://math.stackexchange.com/q/2949131/121988">Symmetric Brick Stacking</a>, Mathematics Stack Exchange, 2018.
%H Doron Zeilberger, <a href="https://arxiv.org/abs/1208.2258">The amazing 3^n theorem and its even more amazing proof</a>, arXiv:1208.2258 [math.CO], 2012.
%H Doron Zeilberger, <a href="/A333650/a333650.jpg">The 27 towers with 4 domino pieces</a>, illustration.
%F Row sums are given by A000244(n-1) = 3^(n-1).
%F T(n,1) = 1.
%F T(n,n) = 2^(n-1).
%e Table begins:
%e n\k| 1 2 3 4 5 6 7 8 9 10 11
%e ---+-----------------------------------------------------
%e 1 | 1
%e 2 | 1 2
%e 3 | 1 4 4
%e 4 | 1 7 11 8
%e 5 | 1 12 24 28 16
%e 6 | 1 20 52 70 68 32
%e 7 | 1 33 110 168 193 160 64
%e 8 | 1 54 228 401 497 510 368 128
%e 9 | 1 88 467 944 1257 1412 1304 832 256
%e 10 | 1 143 949 2187 3172 3736 3879 3248 1856 512
%e 11 | 1 232 1916 5010 7946 9778 10766 10360 7920 4096 1024
%e .
%e T(3,2) = 4 because there are four domino towers of height two consisting of three bricks:
%e +-------+-------+ +-------+ +-------+
%e | | | | | | |
%e +---+---+---+---+, +---+---+---+---+, +-------+---+---+---+, and
%e | | | | | | | |
%e +-------+ +-------+-------+ +-------+-------+
%e +-------+
%e | |
%e +---+---+---+-------+.
%e | | |
%e +-------+-------+
%Y Cf. A000071 (col. 2), A339493 (col. 3), A000244, A038622, A168368, A264746, A320314, A339252, A339254, A339029, A339346, A339494.
%K nonn,tabl,hard
%O 1,3
%A _Peter Kagey_, Mar 31 2020
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