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A035175
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Coefficients in expansion of Dirichlet series Product_p (1-(Kronecker(m,p)+1)*p^(-s)+Kronecker(m,p)*p^(-2s))^(-1) for m = -15.
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26
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1, 2, 1, 3, 1, 2, 0, 4, 1, 2, 0, 3, 0, 0, 1, 5, 2, 2, 2, 3, 0, 0, 2, 4, 1, 0, 1, 0, 0, 2, 2, 6, 0, 4, 0, 3, 0, 4, 0, 4, 0, 0, 0, 0, 1, 4, 2, 5, 1, 2, 2, 0, 2, 2, 0, 0, 2, 0, 0, 3, 2, 4, 0, 7, 0, 0, 0, 6, 2, 0, 0, 4, 0, 0, 1, 6, 0, 0, 2, 5, 1, 0, 2, 0, 2, 0, 0, 0, 0, 2, 0, 6, 2, 4, 2, 6, 0, 2, 0, 3, 0, 4, 0, 0, 0
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OFFSET
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1,2
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COMMENTS
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Coefficients of Dedekind zeta function for the quadratic number field of discriminant -15. See A002324 for formula and Maple code. - N. J. A. Sloane, Mar 22 2022
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LINKS
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FORMULA
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Expansion of -1 + (eta(q^3) * eta(q^5))^2 / (eta(q) * eta(q^15)) in powers of q.
Euler transform of period 15 sequence [ 1, 1, -1, 1, -1, -1, 1, 1, -1, -1, 1, -1, 1, 1, -2, ...]. if a(0)=1.
Moebius transform is period 15 sequence [ 1, 1, 0, 1, 0, 0, -1, 1, 0, 0, -1, 0, -1, -1, 0, ...].
Given g.f. A(x), then B(x) = 1 + A(x) satisfies 0 = f(B(x), B(x^2), B(x^4)) where f(u, v, w) = -v^3 + 4*u*v*w - 2*u*w^2 - u^2*w.
G.f.: -1 + x * Product_{k>0} ((1 - x^(3*k)) * (1 - x^(5*k)))^2 / ((1 - x^k) * (1 - x^(15*k))).
G.f.: -1 + (1/2) * (Sum_{n,m} x^(n^2 + n*m + 4*m^2) + x^(2*n^2 + n*m + 2*m^2)).
a(n) is multiplicative with a(3^e) = a(5^e) = 1, a(p^e) = (1+(-1)^e)/2 if p == 7, 11, 13, 14 (mod 15), a(p^e) = e+1 if p == 1, 2, 4, 8 (mod 15).
a(15*n + 7) = a(15*n + 11) = a(15*n + 13) = a(15*n + 14) = 0.
a(3*n) = a(n). a(n) = |A106406(n)| unless n=0. a(n) = A123864(n) unless n=0. (End)
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 2*Pi/sqrt(15) = 1.622311... . - Amiram Eldar, Oct 11 2022
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EXAMPLE
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q + 2*q^2 + q^3 + 3*q^4 + q^5 + 2*q^6 + 4*q^8 + q^9 + 2*q^10 +...
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MATHEMATICA
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QP = QPochhammer; s = (QP[q^3]*QP[q^5])^2/(QP[q]*QP[q^15])/q - 1/q + O[q]^105; CoefficientList[s, q] (* Jean-François Alcover, Nov 30 2015, adapted from PARI *)
a[n_] := If[n < 0, 0, DivisorSum[n, KroneckerSymbol[-15, #] &]]; Table[a[n], {n, 1, 100}] (* G. C. Greubel, Jul 17 2018 *)
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PROG
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(PARI) m = -15; direuler(p=2, 101, 1/(1-(kronecker(m, p)*(X-X^2))-X))
(PARI) {a(n)=if(n<1, 0, sumdiv(n, d, kronecker(-15, d)))} \\ Michael Somos, Aug 25 2006
(PARI) {a(n)=local(A, p, e); if(n<1, 0, A=factor(n); prod(k=1, matsize(A)[1], if(p=A[k, 1], e=A[k, 2]; if(p==3|p==5, 1, if((p%15)!=2^valuation(p%15, 2), (e+1)%2, (e+1))))))} \\ Michael Somos, Aug 25 2006
(PARI) {a(n)=if(n<1, 0, (qfrep([2, 1; 1, 8], n, 1)+qfrep([4, 1; 1, 4], n, 1))[n])} \\ Michael Somos, Aug 25 2006
(PARI) {a(n)=local(A); if(n<1, 0, A=x*O(x^n); polcoeff( eta(x^3+A)^2*eta(x^5+A)^2/eta(x+A)/eta(x^15+A), n))} \\ Michael Somos, Aug 25 2006
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CROSSREFS
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Dedekind zeta functions for imaginary quadratic number fields of discriminants -3, -4, -7, -8, -11, -15, -19, -20 are A002324, A002654, A035182, A002325, A035179, A035175, A035171, A035170, respectively.
Dedekind zeta functions for real quadratic number fields of discriminants 5, 8, 12, 13, 17, 21, 24, 28, 29, 33, 37, 40 are A035187, A035185, A035194, A035195, A035199, A035203, A035188, A035210, A035211, A035215, A035219, A035192, respectively.
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KEYWORD
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nonn,mult
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AUTHOR
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STATUS
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approved
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