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

1,2

Links

Table of n, a(n) for n=1..45.

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

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).

Example

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, ...
  ...

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))))));

Prog

(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

Crossrefs

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

Diagonal gives A212800.

Cf. A116469, A173958, A340560.

Sequence in context: A109590 A074935 A320103 * A078239 A083113 A184847

Adjacent sequences: A212793 A212794 A212795 * A212797 A212798 A212799

Keyword

nonn,tabl

Author

N. J. A. Sloane, May 27 2012

Status

approved