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A212796 Square array read by antidiagonals: T(m,n) = number of spanning trees in C_m X C_n.

%I A212796 #31 Feb 16 2025 08:33:17
%S A212796 1,2,2,3,32,3,4,294,294,4,5,2304,11664,2304,5,6,16810,367500,367500,
%T A212796 16810,6,7,117600,10609215,42467328,10609215,117600,7,8,799694,
%U A212796 292626432,4381392020,4381392020,292626432,799694,8,9,5326848,7839321861,428652000000,1562500000000,428652000000,7839321861,5326848,9
%N A212796 Square array read by antidiagonals: T(m,n) = number of spanning trees in C_m X C_n.
%H A212796 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 A212796 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/SpanningTree.html">Spanning Tree</a>
%H A212796 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/TorusGridGraph.html">Torus Grid Graph</a>
%F A212796 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 A212796 Array begins:
%e A212796   1,    2,      3,        4,          5,            6               7, ...
%e A212796   2,   32,    294,     2304,      16810,       117600,         799694, ...
%e A212796   3,  294,  11664,   367500,   10609215,    292626432,     7839321861, ...
%e A212796   4, 2304, 367500, 42467328, 4381392020, 428652000000, 40643137651228, ...
%e A212796   ...
%p A212796 Digits:=200;
%p A212796 T:=(m,n)->round(Re(evalf(simplify(expand(
%p A212796 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 A212796 (PARI) default(realprecision, 120);
%o A212796 {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 A212796 Rows and columns 1..10 give A000027, A212797, A212798, A212799, A358810, A358811, A358812, A358813, A358814, A358815.
%Y A212796 Diagonal gives A212800.
%Y A212796 Cf. A116469, A173958, A340560.
%K A212796 nonn,tabl
%O A212796 1,2
%A A212796 _N. J. A. Sloane_, May 27 2012