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A156645
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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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3
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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
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OFFSET
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0,5
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LINKS
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FORMULA
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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.
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)
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EXAMPLE
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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;
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MATHEMATICA
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(* 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 *)
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PROG
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(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, 2): k in [0..n], n in [0..12]]; // 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, 2) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Jul 03 2021
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CROSSREFS
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KEYWORD
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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