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A123864 Expansion of (eta(q^3) * eta(q^5))^2 / (eta(q) * eta(q^15)) in powers of q.

%I A123864 #32 Aug 06 2025 10:08:27
%S A123864 1,1,2,1,3,1,2,0,4,1,2,0,3,0,0,1,5,2,2,2,3,0,0,2,4,1,0,1,0,0,2,2,6,0,
%T A123864 4,0,3,0,4,0,4,0,0,0,0,1,4,2,5,1,2,2,0,2,2,0,0,2,0,0,3,2,4,0,7,0,0,0,
%U A123864 6,2,0,0,4,0,0,1,6,0,0,2,5,1,0,2,0,2,0,0,0,0,2,0,6,2,4,2,6,0,2,0,3,0,4,0,0
%N A123864 Expansion of (eta(q^3) * eta(q^5))^2 / (eta(q) * eta(q^15)) in powers of q.
%C A123864 Number 31 of the 74 eta-quotients listed in Table I of Martin (1996).
%C A123864 Multiplicative because this sequence is the inverse Moebius transform of a multiplicative sequence Kronecker(-15, n). - _Andrew Howroyd_, Jul 27 2018
%H A123864 Andrew Howroyd, <a href="/A123864/b123864.txt">Table of n, a(n) for n = 0..1000</a>
%H A123864 David Broadhurst and Daniele Dorigoni, <a href="https://arxiv.org/abs/2507.21352">Resurgent Lambert series with characters</a>, arXiv:2507.21352 [math.NT], 2025. See p. 37.
%H A123864 Yves Martin, <a href="http://dx.doi.org/10.1090/S0002-9947-96-01743-6">Multiplicative eta-quotients</a>, Trans. Amer. Math. Soc. 348 (1996), no. 12, 4825-4856, see page 4852 Table I.
%H A123864 Michael Somos, <a href="/A030203/a030203.txt">Index to Yves Martin's list of 74 multiplicative eta-quotients and their A-numbers</a>, 2016.
%F A123864 Euler transform of period 15 sequence [ 1, 1, -1, 1, -1, -1, 1, 1, -1, -1, 1, -1, 1, 1, -2, ...].
%F A123864 Moebius transform is period 15 sequence [ 1, 1, 0, 1, 0, 0, -1, 1, 0, 0, -1, 0, -1, -1, 0, ...].
%F A123864 G.f. A(q) satisfies 0 = f(A(q), A(q^2), A(q^4)) where f(u, v, w) = - v^3 + 4*u*v*w - 2*u*w^2 - u^2*w.
%F A123864 G.f.: Product_{k>0} ((1 - x^(3*k)) * (1 - x^(5*k)))^2 / ((1 - x^k) * (1 - x^(15*k))).
%F A123864 G.f.: (1/2) * (Sum_{n,m in Z} x^(n^2 + n*m + 4*m^2) + x^(2*n^2 + n*m + 2*m^2)).
%F A123864 a(15*n + 7) = a(15*n + 11) = a(15*n + 13) = a(15*n + 14) = 0. a(3*n) = a(n).
%F A123864 a(n) = A035175(n) unless n=0. a(n) = |A106406(n)| unless n=0.
%F A123864 G.f. is a period 1 Fourier series which satisfies f(-1 / (15 t)) = 15^(1/2) (t/i) f(t) where q = exp(2 Pi i t). - _Michael Somos_, Feb 10 2015
%F A123864 a(n) = Sum_{d | n}  Kronecker(-15, d). - _Andrew Howroyd_, Jul 27 2018
%F A123864 From _Amiram Eldar_, Feb 20 2024: (Start)
%F A123864 Multiplicative with a(p^e) = 1 if p = 3 or 5, e + 1 if Kronecker(-15, p) = 1, and 1 - (e mod 2) if Kronecker(-15, p) = -1.
%F A123864 Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 2*Pi/sqrt(15). (End)
%e A123864 G.f. = 1 + q + 2*q^2 + q^3 + 3*q^4 + q^5 + 2*q^6 + 4*q^8 + q^9 + 2*q^10 + ...
%t A123864 a[ n_] := SeriesCoefficient[ (QPochhammer[ q^3] QPochhammer[ q^5])^2 / ( QPochhammer[ q] QPochhammer[ q^15]), {q, 0, n}]; (* _Michael Somos_, Feb 10 2015 *)
%t A123864 a[ n_] := If[ n < 1, Boole[n == 0], Sum[ KroneckerSymbol[ -15, d], { d, Divisors[ n]}]]; (* _Michael Somos_, Feb 10 2015 *)
%o A123864 (PARI) {a(n) = if( n<1, n==0, sumdiv( n, d, kronecker( -15, d)))};
%o A123864 (PARI) {a(n) = if( n<1, n==0, (qfrep( [2, 1; 1, 8],n, 1) + qfrep( [4, 1; 1, 4], n, 1))[n])};
%o A123864 (PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x^3 + A) * eta(x^5 + A))^2 / (eta(x + A) * eta(x^15 + A)), n))};
%o A123864 (Magma) A := Basis( ModularForms( Gamma1(15), 1), 106); A[1] + A[2] + 2*A[3] + A[4] + 3*A[5] + A[6] + 2*A[7]; /* _Michael Somos_, Feb 10 2015 */
%Y A123864 Cf. A035175, A106406.
%K A123864 nonn,mult
%O A123864 0,3
%A A123864 _Michael Somos_, Oct 14 2006