%I #10 Feb 16 2025 08:33:26

%S 1,-4,4,2,-4,0,2,0,-4,0,0,8,-2,0,0,0,-4,-8,0,8,0,0,8,0,-2,-4,0,-2,0,0,

%T 0,0,-4,-4,8,0,0,0,8,0,0,-8,0,8,-8,0,0,0,-2,-4,4,4,0,0,-2,0,0,-4,0,8,

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

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

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

%H G. C. Greubel, <a href="/A260486/b260486.txt">Table of n, a(n) for n = 0..1000</a>

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

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

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

%F Euler transform of period 12 sequence [ -4, -2, -2, -2, -4, -3, -4, -2, -2, -2, -4, -2, ...].

%F Convolution of A010815 and A257657.

%e G.f. = 1 - 4*x + 4*x^2 + 2*x^3 - 4*x^4 + 2*x^6 - 4*x^8 + 8*x^11 - 2*x^12 + ...

%t a[ n_] := SeriesCoefficient[ EllipticTheta[ 4, 0, x]^2 EllipticTheta[ 4, 0, x^6] / EllipticTheta[ 4, 0, x^3], {x, 0, n}];

%t a[ n_] := SeriesCoefficient[ QPochhammer[ x, x^2]^2 QPochhammer[ x]^2 QPochhammer[ -x^3] / QPochhammer[ x^3], {x, 0, n}];

%t a[ n_] := If[ n < 1, Boole[n == 0], -4 I^(n-1) Sum[ {1, I, -1/2, I, 1, -I/2}[[Mod[d, 6, 1]]] KroneckerSymbol[ -2, n/d], {d, Divisors[ n]}]];

%t a[n_]:= SeriesCoefficient[EllipticTheta[3, 0, -x]^2* EllipticTheta[3, 0, -x^6]/EllipticTheta[3, 0, -x^3], {x, 0, n}]; Table[a[n], {n,0,50}] (* _G. C. Greubel_, Mar 17 2018 *)

%o (PARI) {a(n) = if( n<1, n==0, -4 * I^(n-1) * sumdiv(n, d, [-I/2, 1, I, -1/2, I, 1][d%6+1] * kronecker(-2, n/d)))};

%o (PARI) {a(n) = my(A, p, e); if( n<1, n==0, A = factor(n); -4 * I^(n-1) * prod(k=1, matsize(A)[1], [p, e] = A[k, ]; if( p==2, I, p==3, 1-e/2, p%8 > 4, !(e%2), e+1)))};

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

%Y Cf. A010815, A257657.

%K sign,changed

%O 0,2

%A _Michael Somos_, Jul 26 2015