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).
Original entry on oeis.org
1, 7, 4961, 371647151, 2952717950351617, 2489597262406609716450871, 222812636926792555435326125877303201, 2116840405025957772469476908228785308996001314527, 2134958300495920487325052422663717579194357002081033470045923329
Offset: 0
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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 *)
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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)))}
A340295
a(n) = 4^(2*n^2) * Product_{1<=j,k<=n} (1 - sin(j*Pi/(2*n+1))^2 * cos(k*Pi/(2*n+1))^2).
Original entry on oeis.org
1, 13, 18281, 2732887529, 43384923739812577, 73125714588602035608260981, 13085551252412040683513520733767180041, 248596840858215958581954513797323868183183928594833
Offset: 0
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Table[Resultant[ChebyshevT[4*n+2, x/2], ChebyshevT[4*n+2, I*x/2], x]^(1/4) / 2^n, {n, 0, 10}] (* Vaclav Kotesovec, Jan 04 2021 *)
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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))*cos(k*Pi/(2*n+1)))^2)))}
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{a(n) = sqrtint(sqrtint(polresultant(polchebyshev(4*n+2, 1, x/2), polchebyshev(4*n+2, 1, I*x/2))))/2^n}
A340430
Square array T(n,k), n >= 0, k >= 0, read by antidiagonals, where T(n,k) = 4^(2*n*k) * Product_{a=1..n} Product_{b=1..k} (1 - cos(a*Pi/(2*n+1))^2 * cos(b*Pi/(2*k+1))^2).
Original entry on oeis.org
1, 1, 1, 1, 15, 1, 1, 209, 209, 1, 1, 2911, 32625, 2911, 1, 1, 40545, 5015009, 5015009, 40545, 1, 1, 564719, 770100001, 8238791743, 770100001, 564719, 1, 1, 7865521, 118247646001, 13441754883649, 13441754883649, 118247646001, 7865521, 1
Offset: 0
Square array begins:
1, 1, 1, 1, 1, ...
1, 15, 209, 2911, 40545, ...
1, 209, 32625, 5015009, 770100001, ...
1, 2911, 5015009, 8238791743, 13441754883649, ...
1, 40545, 770100001, 13441754883649, 230629380093001665, ...
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default(realprecision, 120);
{T(n, k) = round(4^(2*n*k)*prod(a=1, n, prod(b=1, k, 1-(cos(a*Pi/(2*n+1))*cos(b*Pi/(2*k+1)))^2)))}
A071102
Determinant of KK* where K is Kasteleyn-Percus matrix for fool's diamond of order n.
Original entry on oeis.org
1, 2, 15, 384, 32625, 9085440, 8238791743, 24233379889152
Offset: 1
- J. Propp, Enumeration of matchings: problems and progress, pp. 255-291 in L. J. Billera et al., eds, New Perspectives in Algebraic Combinatorics, Cambridge, 1999 (see Problem 27).
Showing 1-4 of 4 results.
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