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%I #34 Jul 11 2019 23:14:24
%S 1,1,2,1,1,3,1,3,1,1,6,5,1,2,4,1,1,9,6,1,5,5,1,1,15,35,28,9,1,1,4,1,1,
%T 21,70,84,45,11,1,7,14,7,1,1,24,26,9,1,2,16,20,8,1,1,36,210,462,495,
%U 286,91,15,1,3,9,6,1,1,45,330,924,1287,1001,455,120,17,1
%N Irregular table read by rows: Let P(n,x) be the (monic) minimal polynomial of 2i*cos(Pi/n), where i = sqrt(-1) is the imaginary unit, then a(n,k) = [x^(2k)] P(n,x), n >= 3.
%C For n >= 3, it is easy to see that [x^(2k+1)] P(n,x) = 0, so they are omitted.
%C Row n (n >= 3) has length A023022(n) + 1 = phi(n)/2 + 1.
%C Let {U(n,x)} be defined as: U(0,x) = 0, U(1,x) = 1, U(n,x) = x*U(n-1,x) + U(n-2,x) for n >= 2, then U(n,x) = Product_{k|n, k>=2} P(k,x) for n > 0, because U(n,x) = Product_{m=1..n-1} (x - 2i*cos(Pi*m/n)) for n > 0.
%F P(n,x) = Product_{0<=m<=n, gcd(m, n)=1} (x - 2i*cos(Pi*m/n)).
%F Equivalently, P(n,x) = Product_{0<=m<=n/2, gcd(m, n)=1} (x^2 + 4*cos(Pi*m/n)) for n != 2. This shows that all terms are positive.
%F P(n,x) = Product_{k|n} U(n/k,x)^mu(k), mu = A008683.
%F Let MPR2(n,x) be the (monic) minimal polynomial of 2*cos(2*Pi/n) as defined in A232624, then: for even n > 2, P(n,x) = MPR2(2n,i*x)*(-1)^A023022(n); for odd n, P(n,x) = MPR2(n,i*x)*MPR2(2n,i*x)*(-1)^A023022(n), i = sqrt(-1).
%F For n > 2, P(n,x) = MPR2(n,-x^2-2)*(-1)^A023022(n).
%F For n > 1, P(n,1) = A061446(n), P(n,2) = A008555(n), P(n,3) = A253807(n), ...
%F For even n > 2, a(n,k) = (-1)^(A023022(n)-k)*A232624(2n,2k).
%e P(1,x) = x^2 + 4;
%e P(2,x) = x;
%e P(3,x) = x^2 + 1;
%e P(4,x) = x^2 + 2;
%e P(5,x) = x^4 + 3x^2 + 1;
%e P(6,x) = x^2 + 3;
%e P(7,x) = x^6 + 5x^4 + 6x^2 + 1;
%e P(8,x) = x^4 + 4x^2 + 2;
%e P(9,x) = x^6 + 6x^4 + 9x^2 + 1;
%e P(10,x) = x^4 + 5x^2 + 5;
%e ...
%t ro[n_] := (P = CoefficientList[p = MinimalPolynomial[2*I*Cos[Pi/n], x], x^2]; P); Flatten[Table[ro[n], {n, 3, 30}]]
%o (PARI) U(n) = sum(i=0, (n-1)/2,binomial(n,2*i+1)*(poly/2)^(n-2*i-1)*((poly^2+4)/4)^i)
%o P(n) = if(n==1, poly^2+4, my(v=divisors(n)); prod(i=1, #v, U(n/v[i])^moebius(v[i])))
%o a(n,k) = polcoeff(P(n),2*k)
%Y Cf. A232624.
%Y Cf. P(n,k): A061446 (k=1), A008555 (k=2), A253807 (k=3);
%Y Cf. also A023022, A008683.
%K nonn,tabf
%O 3,3
%A _Jianing Song_, Jul 08 2019
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