A156647 Square array T(n, k) = Product_{j=1..n} (1 - ChebyshevT(j, k+1)^2) with T(n, 0) = n!, read by antidiagonals.

1, 1, 1, 1, -3, 2, 1, -8, 144, 6, 1, -15, 2304, -97200, 24, 1, -24, 14400, -22579200, 914457600, 120, 1, -35, 57600, -857304000, 7517247897600, -119833267276800, 720, 1, -48, 176400, -13548902400, 3163657512960000, -85018329720343756800, 218719679433615360000, 5040

Offset: 0

Author

Roger L. Bagula, Feb 12 2009

Keywords

• sign
• tabl

Examples

			Square array begins as:
    1,                1,                     1, ...;
    1,               -3,                    -8, ...;
    2,              144,                  2304, ...;
    6,           -97200,             -22579200, ...;
   24,        914457600,         7517247897600, ...;
  120, -119833267276800, -85018329720343756800, ...;
Triangle begins as:
  1;
  1,   1;
  1,  -3,     2;
  1,  -8,   144,          6;
  1, -15,  2304,     -97200,            24;
  1, -24, 14400,  -22579200,     914457600,              120;
  1, -35, 57600, -857304000, 7517247897600, -119833267276800, 720;
		

Links

• G. C. Greubel, Antidiagonal rows n = 0..25, flattened

Crossrefs

Cf. A123583.

Programs

Magma

T:= func< n,k | n eq 0 select 1 else k eq 0 select Factorial(n) else (&*[1 - Evaluate(ChebyshevT(j), k+1)^2 : j in [1..n]]) >;
[T(k, n-k): k in [0..n], n in [0..12]]; // G. C. Greubel, Jul 02 2021

Mathematica

T[n_, k_]= If[k==0, n!, Product[1 - ChebyshevT[j, k+1]^2, {j,n}]];
Table[T[k, n-k], {n,0,12}, {k,0,n}]//Flatten (* modified by G. C. Greubel, Jul 02 2021 *)

Sage

def T(n,k): return factorial(n) if (k==0) else product( 1 - chebyshev_T(j, k+1)^2 for j in (1..n) )
flatten([[T(k, n-k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Jul 02 2021

Formula

T(n, k) = Product_{j=1..n} (1 - ChebyshevT(j, k+1)^2) with T(n, 0) = n! (square array).

Extensions

Edited by G. C. Greubel, Jul 02 2021