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A348566 Triangle read by rows: T(m, n) is the number of symmetric recurrent sandpiles on an m X n grid (m >= 0, 0 <= n <= m).

%I A348566 #7 Oct 22 2021 23:47:59
%S A348566 1,1,4,1,3,2,1,14,7,128,1,11,5,71,36,1,52,18,1358,539,43264,1,41,13,
%T A348566 769,281,17753,6728,1,194,47,14852,4271,1452866,434657,151519232,1,
%U A348566 153,34,8449,2245,603126,167089,46069729,12988816,1,724,123,163534,34276,49704772,10894561,16236962114,3625549353,5475450241024
%N A348566 Triangle read by rows: T(m, n) is the number of symmetric recurrent sandpiles on an m X n grid (m >= 0, 0 <= n <= m).
%C A348566 Terms of this triangle count recurrent sandpiles on rectangular grids that have vertical and horizontal symmetries. Terms of A348567 count recurrent sandpiles on square grids that also have diagonal symmetries.
%H A348566 Laura Florescu, Daniela Morar, David Perkinson, Nicholas Salter and Tianyuan Xu, <a href="https://doi.org/10.37236/4472">Sandpiles and Dominos</a>, El. J. Comb., 22 (2015), P1.66.
%H A348566 Wikipedia, <a href="https://en.wikipedia.org/wiki/Abelian_sandpile_model">Abelian sandpile model</a>
%F A348566 T(2m, 2n) = A187617(m, n) = A187618(m, n). [Florescu et al., Theorem 15]
%F A348566 T(2m, 2n-1) = T(2n-1, 2m) = A103997(m, n). [Florescu et al., Theorem 18]
%F A348566 T(2m-1, 2n-1) = Product_{h=1..m, k=1..n} 4*(z(h, m) + z(k, n)) where z(k, n) = cos(Pi*(2k-1)/(4n)). [Florescu et al., Theorem 23]
%F A348566 A256045(m, n) divides T(m, n), T(m, n) divides A116469(m+1, n+1).
%F A348566 This triangle can obviously be extended to n > m as T(m, n) = T(n, m).
%e A348566 The triangle begins:
%e A348566   1
%e A348566   1  4
%e A348566   1  3  2
%e A348566   1 14  7  128
%e A348566   1 11  5   71  36
%e A348566   1 52 18 1358 539 43264
%e A348566   1 41 13  769 281 17753 6728
%e A348566 ...
%e A348566 See Fig. 9 of the paper by Florescu et al. for the T(4, 4) = 36 symmetric recurrent sandpiles on a 4x4 grid.
%Y A348566 Cf. A348567, A187617, A187618, A103997, A256045, A116469.
%K A348566 nonn,tabl
%O A348566 0,3
%A A348566 _Andrey Zabolotskiy_, Oct 22 2021