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

Author

N. J. A. Sloane, May 27 2012

Keywords

• nonn
• tabl

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

Rows and columns 1..10 give A000027, A212797, A212798, A212799, A358810, A358811, A358812, A358813, A358814, A358815.

Diagonal gives A212800.

Cf. A116469, A173958, A340560.

Programs

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))))));

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).