A334178 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).

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, 18, 176, 769, 436, 571, 1, 128, 1, 29, 539, 2911, 8449, 2089, 2131, 1, 256, 1, 47, 1471, 17753, 48301, 93127, 10009, 7953, 1, 512, 1, 76, 4271, 79808, 603126, 801701, 1027207, 47956, 29681, 1, 1024

Offset: 0

Author

Seiichi Manyama, Apr 17 2020

Keywords

• nonn
• tabl

Examples

			Square array begins:
   1, 1,    1,     1,       1,        1,         1, ...
   2, 1,    3,     4,       7,       11,        18, ...
   4, 1,   11,    19,      71,      176,       539, ...
   8, 1,   41,    91,     769,     2911,     17753, ...
  16, 1,  153,   436,    8449,    48301,    603126, ...
  32, 1,  571,  2089,   93127,   801701,  20721019, ...
  64, 1, 2131, 10009, 1027207, 13307111, 714790675, ...
		

Crossrefs

Rows 0..1 give A000012, A000032.

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

Cf. A103997.

Programs

Mathematica

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 *)

PARI

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

Formula

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