A340293 -id:A340293 - cp's OEIS frontend

A340292 a(n) = 4^(2n^2) Product_{1<=j,k<=n} (1 - sin(jPi/(2n+1))^2 * sin(kPi/(2n+1))^2).

1, 7, 4961, 371647151, 2952717950351617, 2489597262406609716450871, 222812636926792555435326125877303201, 2116840405025957772469476908228785308996001314527, 2134958300495920487325052422663717579194357002081033470045923329

Offset: 0

Seiichi Manyama, Jan 03 2021

nonn

Cf. A002315, A340166, A340291, A340293, A340295.

Table[2^(4*n^2) * Product[Product[1 - Sin[j*Pi/(2*n + 1)]^2 * Sin[k*Pi/(2*n + 1)]^2, {k, 1, n}], {j, 1, n}], {n, 0, 10}] // Round (* Vaclav Kotesovec, Jan 04 2021 *)

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

a(n) = A002315(n) * A340293(n)^2.

a(n) ~ exp(2*G*(2*n+1)^2/Pi) / 2^(4*n + 3/4), where G is Catalan's constant A006752. - Vaclav Kotesovec, Jan 04 2021

A340396 a(n) = 2^(n^2 - 1) * Product_{j=1..n, k=1..n} (1 + sin(Pij/n)^2 + sin(Pik/n)^2).

Original entry on oeis.org

0, 1, 96, 93789, 1244160000, 241885578271872, 700566272328037500000, 30323548995402141685610526683, 19627362048402730985830806120284160000, 189995156103157091521654945902925881881155376920, 27506190205802587152768139358989866456457087869970721213256

Offset: 0

Vaclav Kotesovec, Jan 06 2021

nonn

Cf. A004003, A007726, A065072, A067518, A127605, A340052, A340165, A340182, A340183, A340293.

Table[2^(n^2 - 1) * Product[1 + Sin[Pi*j/n]^2 + Sin[Pi*k/n]^2, {j, 1, n}, {k, 1, n}], {n, 0, 10}] // Round

a(n) = 2^(n^2-1) * Product_{j=1..n, k=1..n} (3 - cos(Pi*j/n)^2 - cos(Pi*k/n)^2).
a(n) = 2^(n^2-1) * Product_{j=1..n, k=1..n} (2-cos(2*Pi*j/n)/2-cos(2*Pi*k/n)/2).
a(n) ~ 2^(n^2-1) * exp(4*c*n^2/Pi^2), where c = Integral_{x=0..Pi/2, y=0..Pi/2} log(1 + sin(x)^2 + sin(y)^2) dy dx = -Pi^2*(log(2) + log(sqrt(2)-1)/2) + Pi * Integral_{x=0..Pi/2} log(1 + sqrt(1 + 1/(1 + sin(x)^2))) dx = A340421 = 1.627008991085721315763766677017604437985734719035793082916212355323520649...

cp's OEIS Frontend

A340292 a(n) = 4^(2n^2) Product_{1<=j,k<=n} (1 - sin(jPi/(2n+1))^2 * sin(kPi/(2n+1))^2).

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Formula

A340396 a(n) = 2^(n^2 - 1) * Product_{j=1..n, k=1..n} (1 + sin(Pij/n)^2 + sin(Pik/n)^2).

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cp's OEIS Frontend

A340292 a(n) = 4^(2*n^2) * Product_{1<=j,k<=n} (1 - sin(j*Pi/(2*n+1))^2 * sin(k*Pi/(2*n+1))^2).

Views

Author

Keywords

Crossrefs

Programs

Mathematica

PARI

Formula

A340396 a(n) = 2^(n^2 - 1) * Product_{j=1..n, k=1..n} (1 + sin(Pi*j/n)^2 + sin(Pi*k/n)^2).

Views

Author

Keywords

Crossrefs

Programs

Mathematica

Formula

A340292 a(n) = 4^(2n^2) Product_{1<=j,k<=n} (1 - sin(jPi/(2n+1))^2 * sin(kPi/(2n+1))^2).

A340396 a(n) = 2^(n^2 - 1) * Product_{j=1..n, k=1..n} (1 + sin(Pij/n)^2 + sin(Pik/n)^2).