A256004 - OEIS

%I #13 Sep 08 2022 08:46:11

%S 1,-3,3,0,0,-3,1,3,0,-2,-3,3,3,-3,0,0,0,0,2,-3,3,-3,3,-3,1,0,0,3,-3,0,

%T 0,0,3,-2,0,0,-1,-3,3,3,0,-6,3,3,0,-2,-3,6,0,-6,0,0,3,0,2,-3,3,0,0,-3,

%U 2,0,0,-3,-3,0,3,6,0,-3,0,0,4,-6,3,-3,0,-3,1

%N Expansion of q^3 * f( -q, -q^8)^4 * f( -q^2, -q^7) / (f( -q) * f( -q^4, -q^5)^2) in powers of q where f() is Ramanujan's general 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="/A256004/b256004.txt">Table of n, a(n) for n = 3..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="http://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>

%F Euler transform of period 9 sequence [ -3, 0, 1, 3, 3, 1, 0, -3, -2, ...].

%e G.f. = q^3 - 3*q^4 + 3*q^5 - 3*q^8 + q^9 + 3*q^10 - 2*q^12 - 3*q^13 + ...

%t a[ n_] := If[ n < 3, 0, With[{m = n - 3}, SeriesCoefficient[ q^3 Product[ (1 - q^k)^{3, 0, -1, -3, -3, -1, 0, 3, 2}[[Mod[k, 9, 1]]], {k, m}], {q, 0, n}]]];

%o (PARI) {a(n) = if( n<3, 0, n-=3; polcoeff( prod(k=1, n, (1 - x^k + x * O(x^n))^[2, 3, 0, -1, -3, -3, -1, 0, 3][k%9 + 1]), n))};

%o (Magma) Basis( ModularForms( Gamma1(9), 1), 82) [4];

%K sign

%O 3,2

%A _Michael Somos_, May 06 2015