cp's OEIS Frontend

A133691 Expansion of (1 - (phi(-q) * phi(q^2))^2) / 4 in powers of q where phi() is a Ramanujan theta function.

%I A133691 #19 Feb 16 2025 08:33:06
%S A133691 1,-2,4,-6,6,-8,8,-6,13,-12,12,-24,14,-16,24,-6,18,-26,20,-36,32,-24,
%T A133691 24,-24,31,-28,40,-48,30,-48,32,-6,48,-36,48,-78,38,-40,56,-36,42,-64,
%U A133691 44,-72,78,-48,48,-24,57,-62,72,-84,54,-80,72,-48,80,-60,60,-144
%N A133691 Expansion of (1 - (phi(-q) * phi(q^2))^2) / 4 in powers of q where phi() is a Ramanujan theta function.
%C A133691 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H A133691 G. C. Greubel, <a href="/A133691/b133691.txt">Table of n, a(n) for n = 1..10000</a>
%H A133691 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>.
%H A133691 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>.
%F A133691 Expansion of (1 - (eta(q)^2 * eta(q^4)^5 / (eta(q^2)^3 * eta(q^8)^2))^2) / 4 in powers of q.
%F A133691 a(n) is multiplicative with a(2) = -2, a(2^e) = -6 if e>1, a(p^e) = (p^(e+1) - 1) / (p - 1) if p>2.
%F A133691 a(n) = -4 * A133690(n) = -(-1)^n * A111973(n). a(2*n) = -2 * A046897(n). a(2*n + 1) = A008438(n). a(4*n) = -6 * A000593(n). a(4*n + 1) = A112610(n). a(4*n + 3) = 4 * A097723(n).
%F A133691 Dirichlet g.f.: zeta(s) * zeta(s-1) * (1 - 5/2^s + 1/2^(2*s-1) + 1/2^(3*s-3)). - _Amiram Eldar_, Oct 28 2023
%e A133691 G.f. = q - 2*q^2 + 4*q^3 - 6*q^4 + 6*q^5 - 8*q^6 + 8*q^7 - 6*q^8 + 13*q^9 - ...
%t A133691 a[ n_] := Which[ n < 1, 0, OddQ[n], DivisorSigma[ 1, n], True, -2 DivisorSum[ n/2, # Boole[Mod[#, 4] > 0] &]]; (* _Michael Somos_, Oct 30 2015 *)
%t A133691 a[ n_] := SeriesCoefficient[ (1 - (EllipticTheta[ 4, 0, q] EllipticTheta[ 3, 0, q^2])^2) / 4, {q, 0, n}]; (* _Michael Somos_, Oct 30 2015 *)
%o A133691 (PARI) {a(n) = if( n<1, 0, n%2, sigma(n), -2 * sumdiv(n/2, d, if(d%4, d)))};
%Y A133691 Cf. A000593, A008438, A046897, A097723, A111973, A112610, A133690.
%K A133691 sign,easy,mult
%O A133691 1,2
%A A133691 _Michael Somos_, Sep 20 2007