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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.

%I A322836 #68 Mar 05 2021 09:36:20
%S A322836 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,
%T A322836 49,244,577,362,1,0,1,7,71,485,1921,3363,1351,1,1,1,8,97,846,4801,
%U A322836 15124,19601,5042,1,0,1,9,127,1351,10081,47525,119071,114243,18817,1,-1
%N 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.
%H A322836 Seiichi Manyama, <a href="/A322836/b322836.txt">Antidiagonals n = 0..139, flattened</a>
%H A322836 Wikipedia, <a href="https://en.wikipedia.org/wiki/Chebyshev_polynomials">Chebyshev polynomials</a>.
%H A322836 <a href="/index/Ch#Cheby">Index entries for sequences related to Chebyshev polynomials.</a>
%F A322836 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.
%F A322836 A(n,k) = cos(n*arccos(k)). - _Seiichi Manyama_, Mar 05 2021
%F A322836 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
%e A322836 Square array begins:
%e A322836    1, 1,    1,     1,      1,      1,       1, ...
%e A322836    0, 1,    2,     3,      4,      5,       6, ...
%e A322836   -1, 1,    7,    17,     31,     49,      71, ...
%e A322836    0, 1,   26,    99,    244,    485,     846, ...
%e A322836    1, 1,   97,   577,   1921,   4801,   10081, ...
%e A322836    0, 1,  362,  3363,  15124,  47525,  120126, ...
%e A322836   -1, 1, 1351, 19601, 119071, 470449, 1431431, ...
%t A322836 Table[ChebyshevT[n-k, k], {n, 0, 10}, {k, n, 0, -1}] // Flatten (* _Amiram Eldar_, Dec 28 2018 *)
%o A322836 (PARI) T(n,k) = polchebyshev(n,1,k);
%o A322836 matrix(7, 7, n, k, T(n-1,k-1)) \\ _Michel Marcus_, Dec 28 2018
%o A322836 (PARI) T(n, k) = round(cos(n*acos(k)));\\ _Seiichi Manyama_, Mar 05 2021
%o A322836 (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
%Y A322836 Mirror of A101124.
%Y A322836 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.
%Y A322836 Rows 0-10 give A000012, A001477, A056220, A144129, A144130, A243131, A243132, A243133, A243134, A243135, A243136.
%Y A322836 Main diagonal gives A115066.
%Y A322836 Cf. A323182 (Chebyshev polynomial of the second kind).
%K A322836 sign,tabl
%O A322836 0,8
%A A322836 _Seiichi Manyama_, Dec 28 2018