%I #19 Mar 14 2023 09:33:46

%S 1,-2,2,-2,2,-4,2,0,2,-2,4,0,2,-4,0,-4,2,-4,2,0,4,0,0,0,2,-6,4,-2,0,

%T -4,4,0,2,0,4,0,2,-4,0,-4,4,-4,0,0,0,-4,0,0,2,-2,6,-4,4,-4,2,0,0,0,4,

%U 0,4,-4,0,0,2,-8,0,0,4,0,0,0,2,-4,4,-6,0,0,4,0

%N Expansion of (phi(q^3) / phi(q)) * phi(-q^2) * phi(-q^6) in powers of q where phi() is a Ramanujan theta function.

%C Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

%D N. J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; p. 85, Equation (32.72).

%H G. C. Greubel, <a href="/A132003/b132003.txt">Table of n, a(n) for n = 0..10000</a>

%H Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>

%F Expansion of eta(q)^2 * eta(q^4) * eta(q^6)^7 / (eta(q^2)^3 * eta(q^3)^2 * eta(q^12)^3) in powers of q.

%F a(n) = -2*b(n) where b() is multiplicative with b(2^e) = 2*0^e - 1, b(3^e) = 1, b(p^e) = e+1 if p == 1 (mod 4), b(p^e) = (1 + (-1)^e)/2 if p == 3 (mod 4).

%F Euler transform of period 12 sequence [-2, 1, 0, 0, -2, -4, -2, 0, 0, 1, -2, -2, ...].

%F G.f.: 1 - 2 * Sum_{k>0} Kronecker(-36, k) * x^k / (1 + x^k).

%F a(n) = - A132004(n) unless n=0.

%F a(2*n) = A122857(n). a(2*n + 1) = -2 * A125079(n). a(3*n) = a(n). a(3*n + 1) = -2 * A258277(n). a(3*n + 2) = 2 * A258278(n). - _Michael Somos_, Nov 01 2015

%F a(12*n + 7) = a(12*n + 11) = 0. a(4*n + 1) = -2 * A008441(n).

%F a(n) = (-1)^n * A122857(n). Expansion of (phi(-q)^2 + phi(-q^3)^2) / 2 in powers of q where phi() is a Ramanujan theta function. - _Michael Somos_, Mar 05 2023

%e G.f. = 1 - 2*x + 2*x^2 - 2*x^3 + 2*x^4 - 4*x^5 + 2*x^6 + 2*x^8 - 2*x^9 + 4*x^10 + ...

%t a[ n_] := If[ n < 1, Boole[n == 0], -2 DivisorSum[ n, (-1)^(n + #) KroneckerSymbol[ -36, #] &]]; (* _Michael Somos_, Nov 01 2015 *)

%t a[ n_] := If[ n < 1, Boole[n == 0], -2 Times @@ (Which[ # < 5, -(-1)^#, Mod[#, 4] == 3, 1 - Mod[#2, 2], True, #2 + 1] & @@@ FactorInteger @ n)]; (* _Michael Somos_, Nov 01 2015 *)

%t a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, q^3] EllipticTheta[ 4, 0, q^2] EllipticTheta[ 4, 0, q^6] / EllipticTheta[ 3, 0, q], {q, 0, n}]; (* _Michael Somos_, Nov 01 2015 *)

%t a[ n_] := SeriesCoefficient[ (EllipticTheta[ 4, 0, q]^2 + EllipticTheta[ 4, 0, q^3]^2) / 2, {q, 0, n}]; (* _Michael Somos_, Mar 05 2023 *)

%o (PARI) {a(n) = if( n<1, n==0, -2 * sumdiv(n, d, (-1)^(n+d) * kronecker(-36, d)))};

%o (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A)^2 * eta(x^4 + A) * eta(x^6 + A)^7 / (eta(x^2 + A)^3 * eta(x^3 + A)^2 * eta(x^12 + A)^3), n))};

%o (PARI) {a(n) = my(A, p, e); if( n<1, n==0, A = factor(n); -2 * prod(k=1, matsize(A)[1], [p, e] = A[k, ]; if( p==3, 1, p==2, -1, p%4==1, e+1, 1-e%2)))};

%Y Cf. A008441, A122857, A125079, A132004.

%K sign

%O 0,2

%A _Michael Somos_, Aug 06 2007