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A134343 Expansion of psi(-x)^2 in powers of x where psi() is a Ramanujan theta function.

%I A134343 #29 Aug 05 2025 10:31:19
%S A134343 1,-2,1,-2,2,0,3,-2,0,-2,2,-2,1,-2,0,-2,4,0,2,0,1,-4,2,0,2,-2,0,-2,2,
%T A134343 -2,1,-4,0,0,2,0,4,-2,2,-2,0,0,3,-2,0,-2,4,0,2,-2,0,-4,0,0,0,-4,3,-2,
%U A134343 2,0,2,-2,0,0,2,-2,4,-2,0,-2,2,0,3,-2,0,0,4,0,2
%N A134343 Expansion of psi(-x)^2 in powers of x where psi() is a Ramanujan theta function.
%C A134343 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%C A134343 Number 57 of the 74 eta-quotients listed in Table I of Martin (1996).
%H A134343 G. C. Greubel, <a href="/A134343/b134343.txt">Table of n, a(n) for n = 0..1000</a>
%H A134343 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 A134343 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 A134343 Michael Somos, <a href="/A030203/a030203.txt">Index to Yves Martin's list of 74 multiplicative eta-quotients and their A-numbers</a>
%H A134343 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A134343 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A134343 Expansion of q^(-1/4) * (eta(q) * eta(q^4) / eta(q^2))^2 in powers of q.
%F A134343 Euler transform of period 4 sequence [ -2, 0, -2, -2, ...].
%F A134343 G.f. is a period 1 Fourier series which satisfies f(-1 / (64 t)) = 8 (t/i) f(t) where q = exp(2 Pi i t).
%F A134343 a(n) = b(4*n + 1) where b() is multiplicative with b(2^e) = 0^e, b(p^e) = (1 + (-1)^e) / 2 if p == 3 (mod 4), b(p^e) = e+1 if p == 1 (mod 8), b(p^e) = (-1)^e * (e+1) if p == 5 (mod 8).
%F A134343 G.f.: (Product_{k>0} (1 - x^k) * (1 + x^(2*k)))^2.
%F A134343 a(9*n + 5) = a(9*n + 8) = 0. a(n) = (-1)^n * A008441(n). a(2*n) = A113407(n). a(2*n + 1) = -2 * A053692(n).
%F A134343 2 * a(n) = A204531(4*n + 1) = - A246950(n). a(4*n) = A246862(n). a(4*n + 2) = A246683(n). - _Michael Somos_, Jun 22 2015
%F A134343 a(4*n + 1) = -2 * A259287(n). a(4*n + 3) = -2 * A259285(n). - _Michael Somos_, Jun 24 2015
%F A134343 Convolution square is A121613. Convolution cube is A213791. Convolution with A000009 is A143379. Convolution with A000143 is A209942. _Michael Somos_, Jun 22 2015
%F A134343 G.f.: Sum_{k>0 odd} (x^k + x^(3*k)) / (1 + x^(4*k)) * (-1)^floor((k+1) / 4). - _Michael Somos_, Jun 22 2015
%F A134343 G.f.: Sum_{k>0 odd} (x^k - x^(3*k)) / (1 + x^(4*k)) * (-1)^floor(k / 4). - _Michael Somos_, Jun 22 2015
%e A134343 G.f. = 1 - 2*x + x^2 - 2*x^3 + 2*x^4 + 3*x^6 - 2*x^7 - 2*x^9 + 2*x^10 + ...
%e A134343 G.f. = q - 2*q^5 + q^9 - 2*q^13 + 2*q^17 + 3*q^25 - 2*q^29 - 2*q^37 + ...
%t A134343 a[ n_] := If[ n < 0, 0, (-1)^n DivisorSum[ 4 n + 1, (-1)^Quotient[#, 2] &]]; (* _Michael Somos_, Jun 22 2015 *)
%t A134343 a[ n_] := SeriesCoefficient[ EllipticTheta[ 2, Pi/4, x^(1/2)]^2 / (2 x^(1/4)), {x, 0, n}]; (* _Michael Somos_, Jun 22 2015 *)
%t A134343 a[ n_] := SeriesCoefficient[(QPochhammer[ x] QPochhammer[ x^4] / QPochhammer[ x^2])^2, {x, 0, n}]; (* _Michael Somos_, Jun 22 2015 *)
%o A134343 (PARI) {a(n) = if( n<0, 0, (-1)^n * sumdiv( 4*n + 1, d, (-1)^(d\2)))};
%o A134343 (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( ( eta(x + A) * eta(x^4 + A) / eta(x^2 + A) )^2, n))};
%o A134343 (Magma) A := Basis( ModularForms( Gamma1(64), 1), 321); A[2] - 2*A[6] + A[10] - 2*A[14] + 2*A[18] + 3*A[26] - 2*A[30] + 2*A[35] - 2*A[36]; /* _Michael Somos_, Jun 22 2015 */;
%Y A134343 Cf. A008441, A053692, A113407, A121613, A143379, A209942, A213791, A246862, A246683, A246950, A259285, A259287.
%K A134343 sign
%O A134343 0,2
%A A134343 _Michael Somos_, Oct 21 2007