cp's OEIS Frontend

A257653 Expansion of f(-x^2)^3 * phi(x^3) / f(-x^6) in powers of x where phi(), f() are Ramanujan theta functions.

%I A257653 #16 Feb 16 2025 08:33:25
%S A257653 1,0,-3,2,0,-6,6,0,-3,12,0,-6,2,0,-12,0,0,-12,18,0,-6,12,0,0,6,0,-18,
%T A257653 14,0,-18,12,0,-3,12,0,-12,12,0,-18,0,0,-24,12,0,-6,36,0,0,2,0,-21,12,
%U A257653 0,-18,42,0,-12,12,0,-18,0,0,-24,0,0,-24,24,0,-12,24,0
%N A257653 Expansion of f(-x^2)^3 * phi(x^3) / f(-x^6) in powers of x where phi(), f() are Ramanujan theta functions.
%C A257653 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H A257653 G. C. Greubel, <a href="/A257653/b257653.txt">Table of n, a(n) for n = 0..1000</a>
%H A257653 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A257653 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A257653 Expansion of eta(q^2)^3 * eta(q^6)^4 / (eta(q^3)^2 * eta(q^12)^2) in powers of q.
%F A257653 Euler transform of period 12 sequence [ 0, -3, 2, -3, 0, -5, 0, -3, 2, -3, 0, -3, ...].
%F A257653 G.f. is a period 1 Fourier series which satisfies f(-1 / (36 t)) = 18^(3/2) (t/i)^(3/2) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A257651.
%F A257653 G.f.: Product_{k>0} (1 - x^(2*k))^3 * (1 + x^(3*k))^2 / (1 + x^(6*k))^2.
%F A257653 a(3*n) = A014452(n). a(3*n + 1) = 0. a(3*n + 2) = -3 * A257651(n).
%e A257653 G.f. = 1 - 3*x^2 + 2*x^3 - 6*x^5 + 6*x^6 - 3*x^8 + 12*x^9 - 6*x^11 + ...
%t A257653 a[ n_] := SeriesCoefficient[ QPochhammer[ x^2]^3 EllipticTheta[ 3, 0, x^3] / QPochhammer[ x^6], {x, 0, n}];
%o A257653 (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^3 * eta(x^6 + A)^4 / (eta(x^3 + A)^2 * eta(x^12 + A)^2), n))};
%o A257653 (Magma) A := Basis( ModularForms( Gamma0(36), 3/2), 71); A[1] - 3*A[3] + 2*A[4] + 6*A[6];
%Y A257653 Cf. A014452, A257651.
%K A257653 sign
%O A257653 0,3
%A A257653 _Michael Somos_, Jul 25 2015