A328788 - OEIS

%I #25 Dec 18 2024 09:26:58

%S 0,0,0,1,-3,0,6,0,-9,4,0,0,3,0,0,6,-21,0,24,0,-18,8,0,0,-3,0,0,13,-24,

%T 0,36,0,-45,12,0,0,21,0,0,14,-54,0,48,0,-36,24,0,0,-15,0,0,18,-42,0,

%U 78,0,-72,20,0,0,18,0,0,32,-93,0,72,0,-54,24,0,0,15

%N Expansion of psi(x^6)^5/psi(-x^3) * (f(-x)/f(-x^4))^3 in powers of x where psi(), f() are Ramanujan theta functions.

%C Number 125 of the 126 eta-quotients listed in Table 1 of Williams 2012.

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

%C G.f. is a period 1 Fourier series which satisfies f(-1 / (12 t)) = 144 (t/i)^2 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A329651.

%H Antti Karttunen, <a href="/A328788/b328788.txt">Table of n, a(n) for n = 0..16383</a>

%H Antti Karttunen, <a href="/A328788/a328788.txt">Data supplement: n, a(n) computed for n = 0..65537</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>

%H K. S. Williams, <a href="http://dx.doi.org/10.1142/S1793042112500595">Fourier series of a class of eta quotients</a>, Int. J. Number Theory 8 (2012), no. 4, 993-1004.

%F Euler transform of period 12 sequence [-3, -3, -2, 0, -3, 2, -3, 0, -2, -3, -3, -4, ...].

%F Expansion of phi(-x^3) * f(-x^2, -x^10)^6 / f(x, x^5)^3 in powers of x where phi(), f(,) are Ramanujan theta functions.

%F Expansion of eta(q)^3 * eta(q^12)^9 / (eta(q^3) * eta(q^4)^3 * eta(q^6)^4) in powers of q.

%F G.f.: x^3 * Product_{n>=1} (1 - x^(3*n))^4 * (1 + x^n)^2 * (1 + x^(2*n))^6 * (1 - x^n + x^(2*n))^5 * (1 - x^(2*n) + x^(4*n))^9.

%F a(n) = s(n/3) - 3*s(n/4) + 3*s(n/6) - s(n/12) if n>0 where s(x) = sum of divisors of x for integer x else 0.

%F a(2*n + 1) = -3 * A229615(n). a(6*n + 1) = a(6*n + 5) = 0. a(6*n + 3) = A008438(n).

%e G.f. = x^3 - 3*x^4 + 6*x^6 - 9*x^8 + 4*x^9 + 3*x^12 + 6*x^15 + ...

%t a[ n_] := SeriesCoefficient[ 2^(-9/2) x^(-15/4) (EllipticTheta[ 2, 0, x^6]^5 / EllipticTheta[ 2, Pi/4, x^3]) (QPochhammer[ x^2] / QPochhammer[ x^8])^3 ,{x,0,n}] // PowerExpand;

%o (PARI) {a(n) = my(s = x -> if(frac(x), 0, sigma(x))); if( n<3, 0, s(n/3) - 3*s(n/4) + 3*s(n/6) - s(n/12))};

%o (PARI) {a(n) = my(A); n-=3; if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A)^3 * eta(x^12 + A)^9 / (eta(x^3 + A) * eta(x^4 + A)^3 * eta(x^6 + A)^4), n))};

%o (Magma) A := Basis( ModularForms( Gamma0(12), 2), 72); A[4] - 3*A[5];

%Y Cf. A000203, A008438, A229615, A329651.

%K sign

%O 0,5

%A _Michael Somos_, Oct 28 2019