This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A035175 #31 Jul 19 2024 03:12:15
%S A035175 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,
%T A035175 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,
%U A035175 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
%N A035175 Coefficients in expansion of Dirichlet series Product_p (1-(Kronecker(m,p)+1)*p^(-s)+Kronecker(m,p)*p^(-2s))^(-1) for m = -15.
%C A035175 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
%H A035175 G. C. Greubel, <a href="/A035175/b035175.txt">Table of n, a(n) for n = 1..10000</a>
%F A035175 From _Michael Somos_, Aug 25 2006: (Start)
%F A035175 Expansion of -1 + (eta(q^3) * eta(q^5))^2 / (eta(q) * eta(q^15)) in powers of q.
%F A035175 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.
%F A035175 Moebius transform is period 15 sequence [ 1, 1, 0, 1, 0, 0, -1, 1, 0, 0, -1, 0, -1, -1, 0, ...].
%F A035175 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.
%F A035175 G.f.: -1 + x * Product_{k>0} ((1 - x^(3*k)) * (1 - x^(5*k)))^2 / ((1 - x^k) * (1 - x^(15*k))).
%F A035175 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)).
%F A035175 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).
%F A035175 a(15*n + 7) = a(15*n + 11) = a(15*n + 13) = a(15*n + 14) = 0.
%F A035175 a(3*n) = a(n). a(n) = |A106406(n)| unless n=0. a(n) = A123864(n) unless n=0. (End)
%F A035175 Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 2*Pi/sqrt(15) = 1.622311... . - _Amiram Eldar_, Oct 11 2022
%e A035175 q + 2*q^2 + q^3 + 3*q^4 + q^5 + 2*q^6 + 4*q^8 + q^9 + 2*q^10 +...
%t A035175 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 *)
%t A035175 a[n_] := If[n < 0, 0, DivisorSum[n, KroneckerSymbol[-15, #] &]]; Table[a[n], {n, 1, 100}] (* _G. C. Greubel_, Jul 17 2018 *)
%o A035175 (PARI) m = -15; direuler(p=2,101,1/(1-(kronecker(m,p)*(X-X^2))-X))
%o A035175 (PARI) {a(n)=if(n<1, 0, sumdiv(n, d, kronecker(-15,d)))} \\ _Michael Somos_, Aug 25 2006
%o A035175 (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
%o A035175 (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
%o A035175 (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
%Y A035175 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.
%Y A035175 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.
%K A035175 nonn,mult
%O A035175 1,2
%A A035175 _N. J. A. Sloane_