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A129449 Expansion of psi(-x) * psi(-x^3) in powers of x where psi() is a Ramanujan theta function.

%I A129449 #28 Aug 05 2025 10:27:57
%S A129449 1,-1,0,-2,1,0,2,0,0,-2,2,0,1,-1,0,-2,0,0,2,-2,0,-2,0,0,3,0,0,0,2,0,2,
%T A129449 -2,0,-2,0,0,2,-1,0,-2,1,0,0,0,0,-4,2,0,2,0,0,-2,0,0,2,-2,0,0,2,0,1,0,
%U A129449 0,-2,2,0,4,0,0,-2,0,0,0,-3,0,-2,0,0,2,0,0,-2,0,0,3,-2,0,-2,0,0,2,-2,0,0,2,0,2,0,0,-2,2,0,0,0,0
%N A129449 Expansion of psi(-x) * psi(-x^3) in powers of x where psi() is a Ramanujan theta function.
%C A129449 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%C A129449 Number 53 of the 74 eta-quotients listed in Table I of Martin (1996).
%H A129449 G. C. Greubel, <a href="/A129449/b129449.txt">Table of n, a(n) for n = 0..1000</a>
%H A129449 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 A129449 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 A129449 Michael Somos, <a href="/A030203/a030203.txt">Index to Yves Martin's list of 74 multiplicative eta-quotients and their A-numbers</a>
%H A129449 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A129449 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A129449 Expansion of q^(-1/2) * eta(q) * eta(q^3) * eta(q^4) * eta(q^12) / (eta(q^2) * eta(q^6)) in powers of q.
%F A129449 Euler transform of period 12 sequence [ -1, 0, -2, -1, -1, 0, -1, -1, -2, 0, -1, -2, ...].
%F A129449 a(n) = b(2*n + 1) where b() is multiplicative with b(2^e) = 0^e, b(3^e) = (-1)^e, b(p^e) = (1 + (-1)^e) / 2 if p == 5, 11 (mod 12), b(p^e) = e+1 if p == 1 (mod 12), b(p^e) = (-1)^e * (e+1) if p == 7 (mod 12).
%F A129449 G.f. is a period 1 Fourier series which satisfies f(-1 / (48 t)) = 48^(1/2) (t/i) f(t) where q = exp(2 Pi i t).
%F A129449 a(n) = (-1)^n * A033762(n). a(2*n) = A112604(n). a(2*n + 1) = -A112605(n). a(3*n) = A129451(n). a(3*n + 1) = -a(n). a(3*n + 2) = 0.
%F A129449 a(4*n) = A112606(n). a(4*n + 1) = - A112608(n). a(4*n + 2) = 2 * A112607(n). a(4*n + 3) = - 2 * A112609(n).
%F A129449 a(6*n) = A123884(n). a(6*n + 3) = -2 * A121361(n).
%e A129449 G.f. = 1 - x - 2*x^3 + x^4 + 2*x^6 - 2*x^9 + 2*x^10 + x^12 - x^13 - 2*x^15 + ...
%e A129449 G.f. = q - q^3 - 2*q^7 + q^9 + 2*q^13 - 2*q^19 + 2*q^21 + q^25 - q^27 + ...
%t A129449 a[ n_] := SeriesCoefficient[ EllipticTheta[ 2, Pi/4, x^(1/2)] EllipticTheta[ 2, Pi/4, x^(3/2)] / (2 x^(1/2)), {x, 0, n}]; (* _Michael Somos_, Jul 09 2015 *)
%t A129449 a[ n_] := With[ {m = 2 n + 1}, If[ m < 1, 0, Sum[ KroneckerSymbol[ 12, d] KroneckerSymbol[ -4, m/d], {d, Divisors[ m]}]]]; (* _Michael Somos_, Jul 09 2015 *)
%o A129449 (PARI) {a(n) = if( n<0, 0, n = 2*n + 1; sumdiv( n, d, kronecker( -4, d) * kronecker( 12, n/d)))};
%o A129449 (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A) * eta(x^3 + A) * eta(x^4 + A) * eta(x^12 + A)/ (eta(x^2 + A) * eta(x^6 + A)), n))};
%Y A129449 Cf. A033762, A112604, A113605, A112606, A112608, A112609, A121361, A123884, A129451.
%K A129449 sign
%O A129449 0,4
%A A129449 _Michael Somos_, Apr 16 2007