A340475 -id:A340475 - cp's OEIS frontend

A341533 Square array T(n,k), n >= 1, k >= 1, read by antidiagonals, where T(n,k) = sqrt( Product_{a=1..n} Product_{b=1..k} (4sin((2a-1)Pi/(2n))^2 + 4sin((2b-1)*Pi/k)^2) ).

2, 8, 2, 14, 36, 2, 36, 50, 200, 2, 82, 256, 224, 1156, 2, 200, 722, 2916, 1058, 6728, 2, 478, 2916, 9922, 38416, 5054, 39204, 2, 1156, 10082, 80000, 155682, 527076, 24200, 228488, 2, 2786, 38416, 401998, 2775556, 2540032, 7311616, 115934, 1331716, 2

Offset: 1

Seiichi Manyama, Feb 13 2021

			Square array begins:
  2,     8,    14,      36,       82,        200, ...
  2,    36,    50,     256,      722,       2916, ...
  2,   200,   224,    2916,     9922,      80000, ...
  2,  1156,  1058,   38416,   155682,    2775556, ...
  2,  6728,  5054,  527076,  2540032,  105125000, ...
  2, 39204, 24200, 7311616, 41934482, 4115479104, ...

Columns 1..7 give A007395, A341543, A231087, A341544, A231485, A341545, A230033.
Main diagonal gives A341535.
Cf. A340475.

default(realprecision, 120);
T(n, k) = round(sqrt(prod(a=1, n, prod(b=1, k, 4*sin((2*a-1)*Pi/(2*n))^2+4*sin((2*b-1)*Pi/k)^2))));

A340560 Square array T(n,k), n >= 1, k >= 1, read by antidiagonals, where T(n,k) = Product_{a=1..n-1} Product_{b=1..k-1} (4sin(aPi/n)^2 + 4sin(bPi/k)^2).

Original entry on oeis.org

1, 1, 1, 1, 8, 1, 1, 49, 49, 1, 1, 288, 1296, 288, 1, 1, 1681, 30625, 30625, 1681, 1, 1, 9800, 707281, 2654208, 707281, 9800, 1, 1, 57121, 16257024, 219069601, 219069601, 16257024, 57121, 1, 1, 332928, 373301041, 17860500000, 62500000000, 17860500000, 373301041, 332928, 1

Offset: 1

Seiichi Manyama, Jan 11 2021

			Square array begins:
  1,    1,      1,         1,           1, ...
  1,    8,     49,       288,        1681, ...
  1,   49,   1296,     30625,      707281, ...
  1,  288,  30625,   2654208,   219069601, ...
  1, 1681, 707281, 219069601, 62500000000, ...

Rows and columns 1..2 give A000012, A001108.
Main diagonal gives A340562.
Cf. A212796, A340475, A340561.

default(realprecision, 120);
{T(n, k) = round(prod(a=1, n-1, prod(b=1, k-1, 4*sin(a*Pi/n)^2+4*sin(b*Pi/k)^2)))}

T(n,k) = T(k,n).

T(n,k) = A212796(n,k)/(n*k).

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} (4sin(aPi/(2n+1))^2 + 4cos(bPi/(2k+1))^2).

Original entry on oeis.org

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

Seiichi Manyama, Jan 09 2021

			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, ...

Wikipedia, Chebyshev polynomials
Wikipedia, Resultant

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.

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)))}

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

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).

cp's OEIS Frontend

A341533 Square array T(n,k), n >= 1, k >= 1, read by antidiagonals, where T(n,k) = sqrt( Product_{a=1..n} Product_{b=1..k} (4sin((2a-1)Pi/(2n))^2 + 4sin((2b-1)*Pi/k)^2) ).

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A340560 Square array T(n,k), n >= 1, k >= 1, read by antidiagonals, where T(n,k) = Product_{a=1..n-1} Product_{b=1..k-1} (4sin(aPi/n)^2 + 4sin(bPi/k)^2).

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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} (4sin(aPi/(2n+1))^2 + 4cos(bPi/(2k+1))^2).

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cp's OEIS Frontend

A341533 Square array T(n,k), n >= 1, k >= 1, read by antidiagonals, where T(n,k) = sqrt( Product_{a=1..n} Product_{b=1..k} (4*sin((2*a-1)*Pi/(2*n))^2 + 4*sin((2*b-1)*Pi/k)^2) ).

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PARI

A340560 Square array T(n,k), n >= 1, k >= 1, read by antidiagonals, where T(n,k) = Product_{a=1..n-1} Product_{b=1..k-1} (4*sin(a*Pi/n)^2 + 4*sin(b*Pi/k)^2).

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PARI

Formula

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).

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Author

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Examples

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Programs

PARI

PARI

Formula

A341533 Square array T(n,k), n >= 1, k >= 1, read by antidiagonals, where T(n,k) = sqrt( Product_{a=1..n} Product_{b=1..k} (4sin((2a-1)Pi/(2n))^2 + 4sin((2b-1)*Pi/k)^2) ).

A340560 Square array T(n,k), n >= 1, k >= 1, read by antidiagonals, where T(n,k) = Product_{a=1..n-1} Product_{b=1..k-1} (4sin(aPi/n)^2 + 4sin(bPi/k)^2).

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} (4sin(aPi/(2n+1))^2 + 4cos(bPi/(2k+1))^2).