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.

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, 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, 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

Offset

1,3

Comments

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

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

References

B. C. Berndt, Ramanujan's Notebooks Part III, Springer-Verlag, see p. 253.

Links

G. C. Greubel, Table of n, a(n) for n = 1..10000

Michael Somos, Introduction to Ramanujan theta functions.

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions.

Formula

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

G.f.: Sum_{k>0} x^k * (1 + x^(2*k)) * (1 + x^(6*k)) / (1 + x^(10*k)). - Michael Somos, Sep 10 2005

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.

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

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.

2*a(n) = A028586(n) + A033718(n) if n>0. - Michael Somos, Oct 21 2006

a(n) = A124233(n) unless n=0. a(n) = |A111949(n)|. a(2*n + 1) = A129390(n). a(4*n + 3) = 2 * A033764(n).

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Pi/sqrt(5) = 1.404962... . - Amiram Eldar, Oct 11 2022

Example

q + q^2 + 2*q^3 + q^4 + q^5 + 2*q^6 + 2*q^7 + q^8 + 3*q^9 + q^10 + ...

Mathematica

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 *)
a[n_] := If[n < 0, 0, DivisorSum[ n, KroneckerSymbol[-20, #] &]]; Table[a[n], {n, 1, 100}] (* G. C. Greubel, Dec 12 2017 *)

Prog

(PARI) direuler(p=2, 101, 1/(1-(kronecker(m, p)*(X-X^2))-X))
(PARI) {a(n) = if( n<1, 0, sumdiv(n, d, kronecker( -20, d)))} \\ Michael Somos, Sep 10 2005
(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
(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

Crossrefs

Cf. A028586, A033718, A033764, A111949, A124233, A129390.

Moebius transform gives A289741.

Cf. A240920 (primes not inert in Q(sqrt(-5))), A139513 (primes decomposing), A003626 (primes remaining inert).

Dedekind zeta functions for imaginary quadratic number fields of discriminants D = -3..-47, -67, -163: A002324, A002654, A035182, A002325, A035179, A035175, A035171, this sequence, A035167, A192013, A035159, A035155, A035151, A035180, A035147, A035143, A318982, A318983.

Dedekind zeta functions for real quadratic number fields of discriminants D = 5..41: A035187, A035185, A035194, A035195, A035199, A035203, A035188, A035210, A035211, A035215, A035219, A035192, A035223.

Sequence in context: A029313 A144001 A124233 * A111949 A143323 A380653

Adjacent sequences: A035167 A035168 A035169 * A035171 A035172 A035173

Keyword

nonn,mult

Author

N. J. A. Sloane

Status

approved