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%I #30 Dec 02 2022 13:28:41
%S 1,2,2,3,32,3,4,294,294,4,5,2304,11664,2304,5,6,16810,367500,367500,
%T 16810,6,7,117600,10609215,42467328,10609215,117600,7,8,799694,
%U 292626432,4381392020,4381392020,292626432,799694,8,9,5326848,7839321861,428652000000,1562500000000,428652000000,7839321861,5326848,9
%N Square array read by antidiagonals: T(m,n) = number of spanning trees in C_m X C_n.
%H Germain Kreweras, <a href="http://dx.doi.org/10.1016/0095-8956(78)90021-7">Complexité et circuits Eulériens dans les sommes tensorielles de graphes</a>, J. Combin. Theory, B 24 (1978), 202-212. See p. 210, Parag. 4.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/SpanningTree.html">Spanning Tree</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/TorusGridGraph.html">Torus Grid Graph</a>
%F T(m,n) = m*n*Prod(Prod( 4*sin(h*Pi/m)^2+4*sin(k*Pi/n)^2, h=1..m-1), k=1..n-1).
%e Array begins:
%e 1, 2, 3, 4, 5, 6 7, ...
%e 2, 32, 294, 2304, 16810, 117600, 799694, ...
%e 3, 294, 11664, 367500, 10609215, 292626432, 7839321861, ...
%e 4, 2304, 367500, 42467328, 4381392020, 428652000000, 40643137651228, ...
%e ...
%p Digits:=200;
%p T:=(m,n)->round(Re(evalf(simplify(expand(
%p m*n*mul(mul( 4*sin(h*Pi/m)^2+4*sin(k*Pi/n)^2, h=1..m-1), k=1..n-1))))));
%o (PARI) default(realprecision, 120);
%o {T(n, k) = round(n*k*prod(a=1, n-1, prod(b=1, k-1, 4*sin(a*Pi/n)^2+4*sin(b*Pi/k)^2)))} \\ _Seiichi Manyama_, Jan 13 2021
%Y Rows and columns 1..10 give A000027, A212797, A212798, A212799, A358810, A358811, A358812, A358813, A358814, A358815.
%Y Diagonal gives A212800.
%Y Cf. A116469, A173958, A340560.
%K nonn,tabl
%O 1,2
%A _N. J. A. Sloane_, May 27 2012
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