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A134177 Expansion of phi(-x^3) * psi(x^4) + x * phi(-x) * psi(x^12) in powers of x where phi(), psi() are Ramanujan theta functions.

%I A134177 #33 Aug 05 2025 10:30:36
%S A134177 1,1,-2,-2,1,2,0,-2,0,0,-2,0,3,1,-2,-2,2,4,0,0,0,0,-2,0,3,0,-2,-4,0,2,
%T A134177 0,-2,0,0,0,0,2,3,-4,-2,1,2,0,-2,0,0,-2,0,2,2,-2,-2,4,2,0,0,0,0,0,0,3,
%U A134177 0,-4,-2,0,2,0,-2,0,0,0,0,4,3,-2,-2,0,4,0
%N A134177 Expansion of phi(-x^3) * psi(x^4) + x * phi(-x) * psi(x^12) in powers of x where phi(), psi() are Ramanujan theta functions.
%C A134177 Number 64 of the 74 eta-quotients listed in Table I of Martin (1996).
%C A134177 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H A134177 Antti Karttunen, <a href="/A134177/b134177.txt">Table of n, a(n) for n = 0..10000</a>
%H A134177 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 A134177 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 A134177 Michael Somos, <a href="/A030203/a030203.txt">Index to Yves Martin's list of 74 multiplicative eta-quotients and their A-numbers</a>
%H A134177 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A134177 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A134177 Expansion of q^(-1/2) * eta(q^2)^4 * eta(q^3) * eta(q^8) * eta(q^12)^4 / ( eta(q) * eta(q^4)^3 * eta(q^6)^3 * eta(q^24) ) in powers of q.
%F A134177 a(n) = b(2*n + 1) where b() is multiplicative with b(2^e) = 0^e, b(3^e) = 1, b(p^e) = e+1 if p == 1, 11 (mod 24), b(p^e) = (e+1) * (-1)^e if p == 5, 7 (mod 24), b(p^e) = (1 + (-1)^e) / 2 if p == 13, 17, 19, 23 (mod 24).
%F A134177 Euler transform of period 24 sequence [ 1, -3, 0, 0, 1, -1, 1, -1, 0, -3, 1, -2, 1, -3, 0, -1, 1, -1, 1, 0, 0, -3, 1, -2, ...].
%F A134177 G.f. is a period 1 Fourier series which satisfies f(-1 / (96 t)) = 96^(1/2) (t/i) f(t) where q = exp(2 Pi i t).
%F A134177 a(12*n + 6) = a(12*n + 8) = a(12*n + 9) = a(12*n + 11) = 0.
%F A134177 G.f.: Sum_{k>=0} a(k) * x^(2*k+1) = Sum_{k>0} Kronecker( -2, k) * (x^k - x^(3*k)) / (1 - x^(2*k) + x^(4*k)).
%F A134177 G.f.: Product_{k>0} (1 + x^k) * (1 - x^(2*k)) * (1 - x^(3*k)) * (1 + x^(6*k)) * (1 - x^(2*k) + x^(4*k))^2 / (1 - x^(4*k) + x^(8*k)).
%F A134177 a(n) = (-1)^n * A128580(n). a(12*n) = A113780(n).
%e A134177 G.f. = 1 + x - 2*x^2 - 2*x^3 + x^4 + 2*x^5 - 2*x^7 - 2*x^10 + 3*x^12 + x^13 - ...
%e A134177 G.f. = q + q^3 - 2*q^5 - 2*q^7 + q^9 + 2*q^11 - 2*q^15 - 2*q^21 + 3*q^25 + q^27 + ...
%t A134177 a[ n_] := If[ n < 0, 0, With[ {m = 2 n + 1}, DivisorSum[ m, KroneckerSymbol[ 12, #] KroneckerSymbol[ -2, m/#] &]]]; (* _Michael Somos_, Jun 24 2015 *)
%t A134177 a[ n_] := SeriesCoefficient[ (EllipticTheta[ 4, 0, x^3] EllipticTheta[ 2, 0, x^2] + EllipticTheta[ 4, 0, x] EllipticTheta[ 2, 0, x^6]) / (2 x^(1/2)), {x, 0, n}]; (* _Michael Somos_, Jun 24 2015 *)
%o A134177 (PARI) {a(n) = if( n<0, 0, n = 2*n + 1; sumdiv(n, d, kronecker( 12, d) * kronecker( -2, n/d)))};
%o A134177 (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^4 * eta(x^3 + A) * eta(x^8 + A) * eta(x^12 + A)^4 / ( eta(x + A) * eta(x^4 + A)^3 * eta(x^6 + A)^3* eta(x^24 + A) ), n))};
%o A134177 (PARI) {a(n) = my(A, p, e); if( n<0, 0, n = 2*n + 1; A = factor(n); prod( k=1, matsize(A)[1], [p, e] = A[k,]; if( p==2, 0, p==3, 1, p%24>12, !(e%2), (e+1) * kronecker(3, p)^e)))};
%Y A134177 Cf. A113780, A128580.
%K A134177 sign
%O A134177 0,3
%A A134177 _Michael Somos_, Oct 11 2007