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A035170 Coefficients in expansion of Dirichlet series Product_p (1-(Kronecker(m,p)+1)*p^(-s)+Kronecker(m,p)*p^(-2s))^(-1) for m = -20.

%I A035170 #31 Feb 16 2025 08:32:37
%S A035170 1,1,2,1,1,2,2,1,3,1,0,2,0,2,2,1,0,3,0,1,4,0,2,2,1,0,4,2,2,2,0,1,0,0,
%T A035170 2,3,0,0,0,1,2,4,2,0,3,2,2,2,3,1,0,0,0,4,0,2,0,2,0,2,2,0,6,1,0,0,2,0,
%U A035170 4,2,0,3,0,0,2,0,0,0,0,1,5,2,2,4,0,2,4,0,2,3,0,2,0,2,0,2,0,3,0,1,2,0,2,0,4
%N A035170 Coefficients in expansion of Dirichlet series Product_p (1-(Kronecker(m,p)+1)*p^(-s)+Kronecker(m,p)*p^(-2s))^(-1) for m = -20.
%C A035170 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%C A035170 Coefficients of Dedekind zeta function for the quadratic number field of discriminant -20. See A002324 for formula and Maple code. - _N. J. A. Sloane_, Mar 22 2022
%D A035170 B. C. Berndt, Ramanujan's Notebooks Part III, Springer-Verlag, see p. 253.
%H A035170 G. C. Greubel, <a href="/A035170/b035170.txt">Table of n, a(n) for n = 1..10000</a>
%H A035170 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>.
%H A035170 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>.
%F A035170 Multiplicative with a(2^e) = a(5^e) = 1, a(p^e) = e+1 if p == 1, 3, 7, 9 (mod 20), a(p^e) = (1+(-1)^e)/2 if p == 11, 13, 17, 19 (mod 20). - _Michael Somos_, Sep 10 2005
%F A035170 G.f.: Sum_{k>0} x^k * (1 + x^(2*k)) * (1 + x^(6*k)) / (1 + x^(10*k)). - _Michael Somos_, Sep 10 2005
%F A035170 a(2*n) = a(5*n) = a(n), a(20*n + 11) = a(20*n + 13) = a(20*n + 17) = a(20*n + 19) = 0.
%F A035170 Moebius transform is period 20 sequence [ 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, -1, 0, -1, 0, 0, 0, -1, 0, -1, 0, ...]. - _Michael Somos_, Oct 21 2006
%F A035170 Expansion of -1 + (phi(q) * phi(q^5) + phi(q^2) * phi(q^10) + 4 * q^3 * psi(q^4)* psi(q^20)) / 2 in powers of q where phi(), psi() are Ramanujan theta functions.
%F A035170 2*a(n) = A028586(n) + A033718(n) if n>0. - _Michael Somos_, Oct 21 2006
%F A035170 a(n) = A124233(n) unless n=0. a(n) = |A111949(n)|. a(2*n + 1) = A129390(n). a(4*n + 3) = 2 * A033764(n).
%F A035170 Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Pi/sqrt(5) = 1.404962... . - _Amiram Eldar_, Oct 11 2022
%e A035170 q + q^2 + 2*q^3 + q^4 + q^5 + 2*q^6 + 2*q^7 + q^8 + 3*q^9 + q^10 + ...
%t A035170 QP = QPochhammer; s = (1/q) * (QP[q^2]*QP[q^4]*QP[q^5]*(QP[q^10] / (QP[q]* QP[q^20]))-1) + O[q]^105; CoefficientList[s, q] (* _Jean-François Alcover_, Dec 04 2015 *)
%t A035170 a[n_] := If[n < 0, 0, DivisorSum[ n, KroneckerSymbol[-20, #] &]]; Table[a[n], {n, 1, 100}] (* _G. C. Greubel_, Dec 12 2017 *)
%o A035170 (PARI) direuler(p=2,101,1/(1-(kronecker(m,p)*(X-X^2))-X))
%o A035170 (PARI) {a(n) = if( n<1, 0, sumdiv(n, d, kronecker( -20, d)))} \\ _Michael Somos_, Sep 10 2005
%o A035170 (PARI) {a(n) = if( n<1, 0, direuler( p=2, n, 1 / (1 - X) / (1 - kronecker( -20, p) * X) )[n])} \\ _Michael Somos_, Sep 10 2005
%o A035170 (PARI) {a(n) = if( n<1, 0, qfrep([1, 0; 0, 5], n)[n] + qfrep([2, 1; 1, 3], n)[n])} \\ _Michael Somos_, Oct 21 2006
%Y A035170 Cf. A028586, A033718, A033764, A111949, A124233, A129390.
%Y A035170 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 A035170 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 A035170 nonn,mult
%O A035170 1,3
%A A035170 _N. J. A. Sloane_