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Square array T(n,k), n >= 0, k >= 0, read by antidiagonals, where T(n,k) = 2^n * sqrt(Resultant(U_{2*n}(x/2), T_{k}(i*x/2))), where T_n(x) is a Chebyshev polynomial of the first kind, U_n(x) is a Chebyshev polynomial of the second kind and i = sqrt(-1).
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%I #23 May 04 2021 02:11:16

%S 1,1,2,1,1,4,1,3,1,8,1,4,11,1,16,1,7,19,41,1,32,1,11,71,91,153,1,64,1,

%T 18,176,769,436,571,1,128,1,29,539,2911,8449,2089,2131,1,256,1,47,

%U 1471,17753,48301,93127,10009,7953,1,512,1,76,4271,79808,603126,801701,1027207,47956,29681,1,1024

%N Square array T(n,k), n >= 0, k >= 0, read by antidiagonals, where T(n,k) = 2^n * sqrt(Resultant(U_{2*n}(x/2), T_{k}(i*x/2))), where T_n(x) is a Chebyshev polynomial of the first kind, U_n(x) is a Chebyshev polynomial of the second kind and i = sqrt(-1).

%F T(n,2*k) = A103997(n,k) for k > 0.

%e Square array begins:

%e 1, 1, 1, 1, 1, 1, 1, ...

%e 2, 1, 3, 4, 7, 11, 18, ...

%e 4, 1, 11, 19, 71, 176, 539, ...

%e 8, 1, 41, 91, 769, 2911, 17753, ...

%e 16, 1, 153, 436, 8449, 48301, 603126, ...

%e 32, 1, 571, 2089, 93127, 801701, 20721019, ...

%e 64, 1, 2131, 10009, 1027207, 13307111, 714790675, ...

%t T[n_, k_] := 2^n * Sqrt[Resultant[ChebyshevU[2*n, x/2], ChebyshevT[k, I*x/2], x]]; Table[T[k, n - k], {n, 0, 10}, {k, 0, n}] // Flatten (* _Amiram Eldar_, May 04 2021 *)

%o (PARI) {T(n, k) = sqrtint(4^n*polresultant(polchebyshev(2*n, 2, x/2), polchebyshev(k, 1, I*x/2)))}

%Y Rows 0..1 give A000012, A000032.

%Y Columns 0..15 give A000079, A000012, A001835(n+1), A004253(n+1), A334135, A003729, A334179, A028478, A334180, A028480, A334181, A028482, A334182, A028484, A334183, A028486.

%Y Cf. A103997.

%K nonn,tabl

%O 0,3

%A _Seiichi Manyama_, Apr 17 2020