A322836 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where A(n,k) is Chebyshev polynomial of the first kind T_{n}(x), evaluated at x=k.
1, 1, 0, 1, 1, -1, 1, 2, 1, 0, 1, 3, 7, 1, 1, 1, 4, 17, 26, 1, 0, 1, 5, 31, 99, 97, 1, -1, 1, 6, 49, 244, 577, 362, 1, 0, 1, 7, 71, 485, 1921, 3363, 1351, 1, 1, 1, 8, 97, 846, 4801, 15124, 19601, 5042, 1, 0, 1, 9, 127, 1351, 10081, 47525, 119071, 114243, 18817, 1, -1
Offset: 0
Examples
Square array begins: 1, 1, 1, 1, 1, 1, 1, ... 0, 1, 2, 3, 4, 5, 6, ... -1, 1, 7, 17, 31, 49, 71, ... 0, 1, 26, 99, 244, 485, 846, ... 1, 1, 97, 577, 1921, 4801, 10081, ... 0, 1, 362, 3363, 15124, 47525, 120126, ... -1, 1, 1351, 19601, 119071, 470449, 1431431, ...
Links
- Seiichi Manyama, Antidiagonals n = 0..139, flattened
- Wikipedia, Chebyshev polynomials.
- Index entries for sequences related to Chebyshev polynomials.
Crossrefs
Mirror of A101124.
Columns 0-20 give A056594, A000012, A001075, A001541, A001091, A001079, A023038, A011943(n+1), A001081, A023039, A001085, A077422, A077424, A097308, A097310, A068203, A322888, A056771, A322889, A078986, A322890.
Rows 0-10 give A000012, A001477, A056220, A144129, A144130, A243131, A243132, A243133, A243134, A243135, A243136.
Main diagonal gives A115066.
Cf. A323182 (Chebyshev polynomial of the second kind).
Programs
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Mathematica
Table[ChebyshevT[n-k, k], {n, 0, 10}, {k, n, 0, -1}] // Flatten (* Amiram Eldar, Dec 28 2018 *) -
PARI
T(n,k) = polchebyshev(n,1,k); matrix(7, 7, n, k, T(n-1,k-1)) \\ Michel Marcus, Dec 28 2018
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PARI
T(n, k) = round(cos(n*acos(k)));\\ Seiichi Manyama, Mar 05 2021
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PARI
T(n, k) = if(n==0, 1, n*sum(j=0, n, (2*k-2)^j*binomial(n+j, 2*j)/(n+j))); \\ Seiichi Manyama, Mar 05 2021
Formula
A(0,k) = 1, A(1,k) = k and A(n,k) = 2 * k * A(n-1,k) - A(n-2,k) for n > 1.
A(n,k) = cos(n*arccos(k)). - Seiichi Manyama, Mar 05 2021
A(n,k) = n * Sum_{j=0..n} (2*k-2)^j * binomial(n+j,2*j)/(n+j) for n > 0. - Seiichi Manyama, Mar 05 2021