A173585 -id:A173585 - cp's OEIS frontend

A156645 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 = 2, read by rows.

1, 1, 1, 1, 36, 1, 1, 1225, 1225, 1, 1, 41616, 1416100, 41616, 1, 1, 1413721, 1634261476, 1634261476, 1413721, 1, 1, 48024900, 1885939157025, 64069586905104, 1885939157025, 48024900, 1, 1, 1631432881, 2176372249076025, 2511659716192658889, 2511659716192658889, 2176372249076025, 1631432881, 1

Offset: 0

Roger L. Bagula, Feb 12 2009

			Triangle begins as:
  1;
  1,        1;
  1,       36,             1;
  1,     1225,          1225,              1;
  1,    41616,       1416100,          41616,             1;
  1,  1413721,    1634261476,     1634261476,       1413721,        1;
  1, 48024900, 1885939157025, 64069586905104, 1885939157025, 48024900, 1;

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

Cf. A007318 (m=0), A173585 (m=1), this sequence (m=2), A156646 (m=10).

Cf. A123583, A156647.

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,2): k in [0..n], n in [0..12]]; // G. C. Greubel, Jul 03 2021

(* 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, 2], {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,2], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Jul 03 2021 *)

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, 2) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Jul 03 2021

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 = 2.
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 = 2.
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 = 2. (End)

Edited by G. C. Greubel, Jul 03 2021

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.

Original entry on oeis.org

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

Roger L. Bagula, Feb 12 2009

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

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

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

Cf. A123583, A156647.

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

(* 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 *)

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

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)

Edited by G. C. Greubel, Jul 03 2021

cp's OEIS Frontend

A156645 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 = 2, read by rows.

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