A156646 Triangle T(n, k, m) = b(n,m)/(b(k,m)*b(n-k,m)), where b(n, k) = Product_{j=1..n} (1 - ChebyshevT(j, k+1)^2), b(n, 0) = n!, and m = 10, read by rows.

1, 1, 1, 1, 484, 1, 1, 233289, 233289, 1, 1, 112444816, 54198633636, 112444816, 1, 1, 54198168025, 12591535188240100, 12591535188240100, 54198168025, 1, 1, 26123404543236, 2925290638056514680225, 1409984043580226203632400, 2925290638056514680225, 26123404543236, 1

Offset: 0

Author

Roger L. Bagula, Feb 12 2009

Keywords

• nonn
• tabl

Examples

			Triangle begins as:
  1;
  1,           1;
  1,         484,                 1;
  1,      233289,            233289,                 1;
  1,   112444816,       54198633636,         112444816,           1;
  1, 54198168025, 12591535188240100, 12591535188240100, 54198168025, 1;
		

Links

• G. C. Greubel, Rows n = 0..25 of the triangle, flattened

Crossrefs

Cf. A007318 (m=0), A173585 (m=1), A156645 (m=2), this sequence (m=10).

Cf. A123583, A156647.

Programs

Magma

b:= 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:= func< n,k,m | b(n,m)/(b(k,m)*b(n-k,m)) >;
[T(n,k,10): k in [0..n], n in [0..12]]; // G. C. Greubel, Jul 03 2021

Mathematica

(* First program *)
b[n_, k_]:= b[n,k]= If[k==0, n!, Product[1 -ChebyshevT[j, k+1]^2, {j,n}]];
T[n_, k_, m_]= b[n,m]/(b[k,m]*b[n-k,m]);
Table[T[n,k,10], {n,0,12}, {k,0,n}]//Flatten (* modified by G. C. Greubel, Jul 03 2021 *)
(* Second program *)
T[n_, k_, m_]:= T[n, k, m]= Product[(1 - ChebyshevT[2*j+2*k, m+1])/(1 - ChebyshevT[2*j, m+1]), {j, n-k}];
Table[T[n,k,12], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Jul 03 2021 *)

Sage

def b(n, k): return factorial(n) if (k==0) else product( 1 - chebyshev_T(j, k+1)^2 for j in (1..n) )
def T(n, k, m): return b(n,m)/(b(k,m)*b(n-k,m))
flatten([[T(n, k, 10) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Jul 03 2021

Formula

T(n, k, m) = b(n,m)/(b(k,m)*b(n-k,m)), where b(n, k) = Product_{j=1..n} (1 - ChebyshevT(j, k+1)^2), b(n, 0) = n!, and m = 10.

From G. C. Greubel, Jul 03 2021: (Start)

T(n, k, m) = b(n,m)/(b(k,m)*b(n-k,m)), where b(n, k) = (1/2^n)*Product_{j=1..n} (1 - ChebyshevT(2*j, k+1)), b(n, 0) = n!, and m = 10.

T(n, k, m) = Product_{j=1..n-k} ( (1 - ChebyshevT(2*j+2*k, m+1))/(1 - ChebyshevT(2*j, m+1)) ) with m = 10. (End)

Extensions

Edited by G. C. Greubel, Jul 03 2021