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%I #30 Oct 11 2022 06:14:08
%S 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 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 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 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 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%C 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 B. C. Berndt, Ramanujan's Notebooks Part III, Springer-Verlag, see p. 253.
%H G. C. Greubel, <a href="/A035170/b035170.txt">Table of n, a(n) for n = 1..10000</a>
%H Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>.
%F 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 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 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 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 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 2*a(n) = A028586(n) + A033718(n) if n>0. - _Michael Somos_, Oct 21 2006
%F 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 Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Pi/sqrt(5) = 1.404962... . - _Amiram Eldar_, Oct 11 2022
%e q + q^2 + 2*q^3 + q^4 + q^5 + 2*q^6 + 2*q^7 + q^8 + 3*q^9 + q^10 + ...
%t 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 a[n_] := If[n < 0, 0, DivisorSum[ n, KroneckerSymbol[-20, #] &]]; Table[a[n], {n, 1, 100}] (* _G. C. Greubel_, Dec 12 2017 *)
%o (PARI) direuler(p=2,101,1/(1-(kronecker(m,p)*(X-X^2))-X))
%o (PARI) {a(n) = if( n<1, 0, sumdiv(n, d, kronecker( -20, d)))} \\ _Michael Somos_, Sep 10 2005
%o (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 (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 Cf. A028586, A033718, A033764, A111949, A124233, A129390.
%Y 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 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 nonn,mult
%O 1,3
%A _N. J. A. Sloane_
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