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A133692 Expansion of phi(-q) * phi(q^2) in powers of q where phi() is a Ramanujan theta function.

%I A133692 #21 Feb 16 2025 08:33:06
%S A133692 1,-2,2,-4,2,0,4,0,2,-6,0,-4,4,0,0,0,2,-4,6,-4,0,0,4,0,4,-2,0,-8,0,0,
%T A133692 0,0,2,-8,4,0,6,0,4,0,0,-4,0,-4,4,0,0,0,4,-2,2,-8,0,0,8,0,0,-8,0,-4,0,
%U A133692 0,0,0,2,0,8,-4,4,0,0,0,6,-4,0,-4,4,0,0,0,0
%N A133692 Expansion of phi(-q) * phi(q^2) in powers of q where phi() is a Ramanujan theta function.
%C A133692 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%C A133692 a(n) is nonzero if and only if n is in A002479.
%H A133692 G. C. Greubel, <a href="/A133692/b133692.txt">Table of n, a(n) for n = 0..5000</a>
%H A133692 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A133692 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A133692 Expansion of eta(q)^2 * eta(q^4)^5 / (eta(q^2)^3 * eta(q^8)^2) in powers of q.
%F A133692 Euler transform of period 8 sequence [ -2, 1, -2, -4, -2, 1, -2, -2, ...].
%F A133692 Moebius transform is period 16 sequence [ -2, 4, -2, 0, 2, 4, 2, 0, -2, -4, -2, 0, 2, -4, 2, 0, ...].
%F A133692 a(n) = -2 * b(n) where b(n) is multiplicative with b(2^e) = -1 if e>0, b(p^e) = (1 + (-1)^e) / 2 if p == 5, 7 (mod 8), b(p^e) = e + 1 if p == 1, 3 (mod 8).
%F A133692 G.f.: Product_{k>0} (1 - x^k)^2 * (1 + x^(2*k))^3 / (1 + x^(4*k))^2.
%F A133692 a(8*n + 5) = a(8*n + 7) = 0.
%F A133692 A133690 is the convolution square. a(n) = (-1)^n * A033715(n). a(2*n) = A033715(n). a(2*n + 1) = -2 * A113411(n).
%e A133692 G.f. = 1 - 2*q + 2*q^2 - 4*q^3 + 2*q^4 + 4*q^6 + 2*q^8 - 6*q^9 - 4*q^11 + ...
%t A133692 a[ n_] := If[ n < 1, Boole[n == 0], (-1)^n 2 DivisorSum[ n, KroneckerSymbol[ -2, #] &]]; (* _Michael Somos_, Oct 30 2015 *)
%t A133692 a[ n_] := SeriesCoefficient[ EllipticTheta[ 4, 0, q] EllipticTheta[ 3, 0, q^2], {q, 0, n}]; (* _Michael Somos_, Aug 29 2014 *)
%o A133692 (PARI) {a(n) = if( n<1, n==0, (-1)^n * 2 * sumdiv(n, d, kronecker( -2, d)))};
%o A133692 (PARI) {a(n) = my(A); if ( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A)^2 * eta(x^4 + A)^5 / (eta(x^2 + A)^3 * eta(x^8 + A)^2), n))};
%o A133692 (Magma) A := Basis( ModularForms( Gamma1(16), 1), 80); A[1] -2*A[2] +2*A[3] - 4*A[4] + 2*A[5] + 4*A[7]; /* _Michael Somos_, Aug 29 2014 */
%Y A133692 Cf. A002479, A033715, A113441, A133690.
%K A133692 sign
%O A133692 0,2
%A A133692 _Michael Somos_, Sep 20 2007