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%I #27 Feb 16 2022 20:57:05
%S 1,1,1,1,6,1,1,16,34,1,1,54,196,198,1,1,196,1666,2704,1154,1,1,726,
%T 16384,64152,37636,6726,1,1,2704,171394,1844164,2549186,524176,39202,
%U 1,1,10086,1844164,57523158,220581904,101757654,7300804,228486,1
%N Square array T(M,N) read by antidiagonals: number of dimer tilings of a 2M x 2N Klein bottle.
%H Cliff, Danny and Zoe Stoll, <a href="http://www.kleinbottle.com">About Klein bottles</a>
%H W. T. Lu and F. Y. Wu, <a href="http://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.
%F T(M, N) = Product_{m=1..M} Product_{n=1..N} ( 4sin(Pi*(4n-1)/(4N))^2 + 4sin(Pi*(2m-1)/(2M))^2 ).
%e Array begins:
%e 1, 1, 1, 1, 1, 1, 1, ...
%e 1, 6, 34, 198, 1154, 6726, 39202, ...
%e 1, 16, 196, 2704, 37636, 524176, 7300804, ...
%e 1, 54, 1666, 64152, 2549186, 101757654, 4064620168, ...
%e 1, 196, 16384, 1844164, 220581904, 26743369156, 3252222705664, ...
%e 1, 726,171394, 57523158, 21050622914, 7902001927776, 2988827208115522, ...
%t T[m_, n_] := Product[4 Sin[(4k-1) Pi/(4n)]^2 + 4 Cos[j Pi/(2m+1)]^2, {j, 1, m}, {k, 1, n}] // Round;
%t Table[T[m-n, n], {m, 0, 9}, {n, 0, m}] // Flatten (* _Jean-François Alcover_, Aug 20 2018 *)
%o (PARI) default(realprecision, 120);
%o {T(n, k) = round(prod(a=1, n, prod(b=1, k, 4*sin((4*a-1)*Pi/(4*n))^2+4*sin((2*b-1)*Pi/(2*k))^2)))} \\ _Seiichi Manyama_, Jan 11 2021
%Y Rows include A003499, A067902+2. Columns include A003500+2.
%Y Main diagonal gives A340557.
%Y Cf. A099390, A103997.
%K nonn,tabl
%O 0,5
%A _Ralf Stephan_, Feb 26 2005
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