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 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. 3
 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 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,5 LINKS G. C. Greubel, Rows n = 0..25 of the triangle, flattened 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 = 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) EXAMPLE 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; 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, 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 *) PROG (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 CROSSREFS Cf. A007318 (m=0), A173585 (m=1), this sequence (m=2), A156646 (m=10). Cf. A123583, A156647. Sequence in context: A181635 A174673 A203277 * A330084 A350385 A374498 Adjacent sequences: A156642 A156643 A156644 * A156646 A156647 A156648 KEYWORD nonn,tabl AUTHOR Roger L. Bagula, Feb 12 2009 EXTENSIONS Edited by G. C. Greubel, Jul 03 2021 STATUS approved

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Last modified July 22 00:08 EDT 2024. Contains 374478 sequences. (Running on oeis4.)