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A209613 Expansion of q * phi(-q^2)^2 * psi(q^3) * psi(-q^3)^2 / psi(q) in powers of q where phi(), psi() are Ramanujan theta functions.

%I A209613 #43 Aug 05 2025 10:33:58
%S A209613 1,-1,-3,1,4,3,-6,-1,9,-4,-12,-3,14,6,-12,1,16,-9,-18,4,18,12,-24,3,
%T A209613 21,-14,-27,-6,28,12,-30,-1,36,-16,-24,9,38,18,-42,-4,40,-18,-42,-12,
%U A209613 36,24,-48,-3,43,-21,-48,14,52,27,-48,6,54,-28,-60,-12,62,30
%N A209613 Expansion of q * phi(-q^2)^2 * psi(q^3) * psi(-q^3)^2 / psi(q) in powers of q where phi(), psi() are Ramanujan theta functions.
%C A209613 Number 27 of the 74 eta-quotients listed in Table I of Martin (1996).
%C A209613 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H A209613 G. C. Greubel, <a href="/A209613/b209613.txt">Table of n, a(n) for n = 1..10000</a>
%H A209613 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. 44.
%H A209613 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 A209613 Michael Somos, <a href="/A030203/a030203.txt">Index to Yves Martin's list of 74 multiplicative eta-quotients and their A-numbers</a>, 2016.
%H A209613 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>, 2019.
%H A209613 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>.
%F A209613 Expansion of eta(q) * eta(q^3) * (eta(q^2) * eta(q^12) / eta(q^4))^2 in powers of q.
%F A209613 Euler transform of period 12 sequence [-1, -3, -2, -1, -1, -4, -1, -1, -2, -3, -1, -4, ...].
%F A209613 a(n) is multiplicative with a(2^e) = (-1)^e, a(3^e) = (-3)^e, a(p^e) = (-1)^(e * (p mod 12 > 6)) * (p^(e+1) - f^(e+1)) / (p - f) if p > 3 where f = Kronecker(3, p).
%F A209613 G.f. is a period 1 Fourier series which satisfies f(-1 / (12 t)) = 192^(1/2) (t/i)^2 g(t) where q = exp(2 Pi i t) and g(t) is g.f. for A113421.
%F A209613 G.f.: Sum_{k>0} k * x^k / (1 + x^k + x^(2*k)) * Kronecker(-4, k).
%F A209613 G.f.: Sum_{k>0} k * x^k / (1 - x^k + x^(2*k)) * A209615(k).
%F A209613 a(2*n) = -a(n) unless n=0. a(3*n) = a(n).
%F A209613 Sum_{k=1..n} abs(a(k)) ~ c * n^2, where c = Pi^2/(18*sqrt(3)) = 0.316567... . - _Amiram Eldar_, Jan 23 2024
%e A209613 G.f. = q - q^2 - 3*q^3 + q^4 + 4*q^5 + 3*q^6 - 6*q^7 - q^8 + 9*q^9 - 4*q^10 + ...
%t A209613 a[ n_] := SeriesCoefficient[ q QPochhammer[ q] QPochhammer[ q^2]^2 QPochhammer[ q^3] QPochhammer[ q^12]^2 / QPochhammer[ q^4]^2 , {q, 0, n}]; (* _Michael Somos_, Jun 09 2015 *)
%t A209613 a[ n_] := If[ n < 1, 0, DivisorSum[ n, # KroneckerSymbol[ -4, #] KroneckerSymbol[ -3, n/#] &]]; (* _Michael Somos_, Jun 09 2015 *)
%o A209613 (PARI) {a(n) = if( n<1, 0, sumdiv( n, d, d * kronecker( -4, d) * kronecker( -3, n/d)))};
%o A209613 (PARI) {a(n) = my(A, p, e, f); if( n<1, 0, A = factor(n); prod( k=1, matsize(A)[1], [p, e] = A[k,]; if( p==2, (-1)^e, p==3, (-3)^e, f = kronecker( 3, p) ; (-1)^(e * (p%12>6)) * (p^(e+1) - f^(e+1)) / (p - f))))};
%o A209613 (PARI) {a(n) = my(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( eta(x + A) * eta(x^3 + A) * (eta(x^2 + A) * eta(x^12 + A) / eta(x^4 + A))^2, n))};
%Y A209613 Cf. A113421, A209615.
%Y A209613 Cf. A000122, A000700, A010054, A121373.
%K A209613 sign,mult
%O A209613 1,3
%A A209613 _Michael Somos_, Mar 10 2012