A212796 Square array read by antidiagonals: T(m,n) = number of spanning trees in C_m X C_n.
1, 2, 2, 3, 32, 3, 4, 294, 294, 4, 5, 2304, 11664, 2304, 5, 6, 16810, 367500, 367500, 16810, 6, 7, 117600, 10609215, 42467328, 10609215, 117600, 7, 8, 799694, 292626432, 4381392020, 4381392020, 292626432, 799694, 8, 9, 5326848, 7839321861, 428652000000, 1562500000000, 428652000000, 7839321861, 5326848, 9
Offset: 1
Examples
Array begins: 1, 2, 3, 4, 5, 6 7, ... 2, 32, 294, 2304, 16810, 117600, 799694, ... 3, 294, 11664, 367500, 10609215, 292626432, 7839321861, ... 4, 2304, 367500, 42467328, 4381392020, 428652000000, 40643137651228, ... ...
Links
- Germain Kreweras, Complexité et circuits Eulériens dans les sommes tensorielles de graphes, J. Combin. Theory, B 24 (1978), 202-212. See p. 210, Parag. 4.
- Eric Weisstein's World of Mathematics, Spanning Tree
- Eric Weisstein's World of Mathematics, Torus Grid Graph
Crossrefs
Programs
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Maple
Digits:=200; T:=(m,n)->round(Re(evalf(simplify(expand( m*n*mul(mul( 4*sin(h*Pi/m)^2+4*sin(k*Pi/n)^2, h=1..m-1), k=1..n-1))))));
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PARI
default(realprecision, 120); {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
Formula
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).