A340427 Square array T(n,k), n >= 1, k >= 1, read by antidiagonals, where T(n,k) = 4^(2*(n-1)*(k-1)) * Product_{a=1..n-1} Product_{b=1..k-1} (1 - sin(a*Pi/(2*n))^2 * sin(b*Pi/(2*k))^2).

1, 1, 1, 1, 12, 1, 1, 140, 140, 1, 1, 1632, 17745, 1632, 1, 1, 19024, 2227120, 2227120, 19024, 1, 1, 221760, 279215849, 2958176256, 279215849, 221760, 1, 1, 2585024, 35001302700, 3909096873216, 3909096873216, 35001302700, 2585024, 1

Offset: 1

Author

Seiichi Manyama, Jan 07 2021

Keywords

• nonn
• tabl

Examples

			Square array begins:
  1,     1,         1,             1,                 1, ...
  1,    12,       140,          1632,             19024, ...
  1,   140,     17745,       2227120,         279215849, ...
  1,  1632,   2227120,    2958176256,     3909096873216, ...
  1, 19024, 279215849, 3909096873216, 54090331699622625, ...
		

Crossrefs

Main diagonal gives A340166.

Cf. A340425, A340428, A340430, A340432.

Programs

PARI

default(realprecision, 120);
{T(n, k) = round(4^(2*(n-1)*(k-1))*prod(a=1, n-1, prod(b=1, k-1, 1-(sin(a*Pi/(2*n))*sin(b*Pi/(2*k)))^2)))}

Formula

T(n,k) = T(k,n).

T(n,k) = 4^(2*(n-1)*(k-1)) * Product_{a=1..n-1} Product_{b=1..k-1} (1 - cos(a*Pi/(2*n))^2 * cos(b*Pi/(2*k))^2).

T(n,k) = 4^(2*(n-1)*(k-1)) * Product_{a=1..n-1} Product_{b=1..k-1} (1 - sin(a*Pi/(2*n))^2 * cos(b*Pi/(2*k))^2).