OFFSET
1,2
COMMENTS
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
LINKS
G. C. Greubel, Table of n, a(n) for n = 1..10000
FORMULA
From Michael Somos, Aug 25 2006: (Start)
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, ...]. [This is A316569. - Jianing Song, Dec 11 2025]
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.
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 2*Pi/sqrt(15) = 1.622311... . - Amiram Eldar, Oct 11 2022
EXAMPLE
q + 2*q^2 + q^3 + 3*q^4 + q^5 + 2*q^6 + 4*q^8 + q^9 + 2*q^10 +...
MATHEMATICA
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 *)
PROG
(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
CROSSREFS
Moebius transform gives A316569.
Dedekind zeta functions for imaginary quadratic number fields of discriminants D = -3..-47, -67, -163: A002324, A002654, A035182, A002325, A035179, this sequence, A035171, A035170, A035167, A192013, A035159, A035155, A035151, A035180, A035147, A035143, A318982, A318983.
KEYWORD
nonn,mult
AUTHOR
STATUS
approved
