A340476 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals, where T(n,k) = Product_{a=1..n} Product_{b=1..k} (4*sin(a*Pi/(2*n+1))^2 + 4*cos(b*Pi/(2*k+1))^2).

1, 1, 1, 1, 4, 1, 1, 19, 11, 1, 1, 91, 176, 29, 1, 1, 436, 2911, 1471, 76, 1, 1, 2089, 48301, 79808, 11989, 199, 1, 1, 10009, 801701, 4375897, 2091817, 97021, 521, 1, 1, 47956, 13307111, 240378643, 372713728, 53924597, 783511, 1364, 1

Offset

0,5

Links

Table of n, a(n) for n=0..44.

Wikipedia, Chebyshev polynomials

Wikipedia, Resultant

Formula

T(n,k) = 2^k * sqrt(Resultant(T_{2*n+1}(i*x/2), U_{2*k}(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).

Example

Square array begins:
  1,  1,     1,       1,         1, ...
  1,  4,    19,      91,       436, ...
  1, 11,   176,    2911,     48301, ...
  1, 29,  1471,   79808,   4375897, ...
  1, 76, 11989, 2091817, 372713728, ...

Prog

(PARI) default(realprecision, 120);
{T(n, k) = round(prod(a=1, n, prod(b=1, k, 4*sin(a*Pi/(2*n+1))^2+4*cos(b*Pi/(2*k+1))^2)))}
(PARI) {T(n, k) = sqrtint(4^k*polresultant(polchebyshev(2*n+1, 1, I*x/2), polchebyshev(2*k, 2, x/2)))}

Crossrefs

Column k=0..1 give A000012, A002878.

Row n=0..7 give A000012, A004253(n+1), A003729, A028478, A028480, A028482, A028484, A028486.

Main diagonal gives A127606.

Cf. A187617, A340475.

Sequence in context: A211709 A323849 A254442 * A176422 A156586 A181544

Adjacent sequences: A340473 A340474 A340475 * A340477 A340478 A340479

Keyword

nonn,tabl

Author

Seiichi Manyama, Jan 09 2021

Status

approved