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A103997 Square array T(M,N) read by antidiagonals: number of dimer tilings of a 2*M X 2*N Moebius strip.

%I A103997 #49 Feb 16 2022 20:40:08
%S A103997 1,1,1,1,3,1,1,11,7,1,1,41,71,18,1,1,153,769,539,47,1,1,571,8449,
%T A103997 17753,4271,123,1,1,2131,93127,603126,434657,34276,322,1,1,7953,
%U A103997 1027207,20721019,46069729,10894561,276119,843,1,1,29681,11332097,714790675,4974089647,3625549353,275770321,2226851,2207,1
%N A103997 Square array T(M,N) read by antidiagonals: number of dimer tilings of a 2*M X 2*N Moebius strip.
%H A103997 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. See Theorem 18.
%H A103997 W. T. Lu and F. Y. Wu, <a href="https://arxiv.org/abs/cond-mat/9906154">Dimer statistics on the Moebius strip and the Klein bottle</a>, arXiv:cond-mat/9906154 [cond-mat.stat-mech], 1999.
%H A103997 <a href="/index/Do#domino">Index entries for sequences related to dominoes</a>
%F A103997 T(M, N) = Product_{m=1..M} (Product_{n=1..N} 4*sin(Pi*(4*n-1)/(4*N))^2 + 4*cos(Pi*m/(2*M + 1))^2).
%F A103997 For k > 0, T(n,k) = 2^n * sqrt(Resultant(U_{2*n}(x/2), T_{2*k}(i*x/2))), where T_n(x) is a Chebyshev polynomial of the first kind, U_n(x) is a Chebyshev polynomial of the second kind and i = sqrt(-1). - _Seiichi Manyama_, Apr 15 2020
%e A103997 Array begins:
%e A103997   1,   1,     1,        1,          1,             1,               1,
%e A103997   1,   3,     7,       18,         47,           123,             322,
%e A103997   1,  11,    71,      539,       4271,         34276,          276119,
%e A103997   1,  41,   769,    17753,     434657,      10894561,       275770321,
%e A103997   1, 153,  8449,   603126,   46069729,    3625549353,    289625349454,
%e A103997   1, 571, 93127, 20721019, 4974089647, 1234496016491, 312007855309063,
%e A103997   ...
%t A103997 T[M_, N_] := Product[4Sin[(4n-1)Pi/(4N)]^2 + 4Cos[m Pi/(2M+1)]^2, {n, 1, N}, {m, 1, M}];
%t A103997 Table[T[M - N, N] // Round, {M, 0, 9}, {N, 0, M}] // Flatten (* _Jean-François Alcover_, Dec 03 2018 *)
%Y A103997 Rows include A005248, A103998.
%Y A103997 Columns 1..7 give A001835(n+1), A334135, A334179, A334180, A334181, A334182, A334183.
%Y A103997 Main diagonal gives A334124.
%Y A103997 Cf. A099390, A103999, A341741, A334178, A256045, A348566.
%K A103997 nonn,tabl
%O A103997 0,5
%A A103997 _Ralf Stephan_, Feb 26 2005