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A133693 Expansion of (1 - phi(-q) * phi(q^2)) / 2 in powers of q where phi() is a Ramanujan theta function.

%I A133693 #19 Feb 16 2025 08:33:06
%S A133693 1,-1,2,-1,0,-2,0,-1,3,0,2,-2,0,0,0,-1,2,-3,2,0,0,-2,0,-2,1,0,4,0,0,0,
%T A133693 0,-1,4,-2,0,-3,0,-2,0,0,2,0,2,-2,0,0,0,-2,1,-1,4,0,0,-4,0,0,4,0,2,0,
%U A133693 0,0,0,-1,0,-4,2,-2,0,0,0,-3,2,0,2,-2,0,0,0,0
%N A133693 Expansion of (1 - phi(-q) * phi(q^2)) / 2 in powers of q where phi() is a Ramanujan theta function.
%C A133693 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%C A133693 For n nonzero, a(n) is nonzero if and only if n is in A002479.
%H A133693 G. C. Greubel, <a href="/A133693/b133693.txt">Table of n, a(n) for n = 1..1000</a>
%H A133693 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A133693 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A133693 Expansion of (1 - eta(q)^2 * eta(q^4)^5 / (eta(q^2)^3 * eta(q^8)^2)) / 2 in powers of q.
%F A133693 Moebius transform is period 16 sequence [ 1, -2, 1, 0, -1, -2, -1, 0, 1, 2, 1, 0, -1, 2, -1, 0, ...].
%F A133693 a(n) is multiplicative with a(2^e) = -1 if e>0, a(p^e) = (1 + (-1)^e) / 2 if p == 5, 7 (mod 8), a(p^e) = e + 1 if p == 1, 3 (mod 8).
%F A133693 a(8*n + 5) = a(8*n + 7) = 0. A133692(n) = -2 * a(n) unless n=0. a(n) = -(-1)^n * A002325(n). a(2*n + 1) = A113411(n).
%e A133693 G.f. = q - q^2 + 2*q^3 - q^4 - 2*q^6 - q^8 + 3*q^9 + 2*q^11 - 2*q^12 - q^16 + ...
%t A133693 a[ n_] := If[ n < 1, 0, -(-1)^n DivisorSum[ n, KroneckerSymbol[ -2, #] &]]; (* _Michael Somos_, Oct 30 2015 *)
%o A133693 (PARI) {a(n) = if( n<1, 0, -(-1)^n * sumdiv(n, d, kronecker( -2, d)))};
%Y A133693 Cf. A002325, A002479, A113411, A133692.
%K A133693 sign,mult
%O A133693 1,3
%A A133693 _Michael Somos_, Sep 20 2007