cp's OEIS Frontend

A209939 Expansion of (f(x) * f(x^3))^3 in powers of q where f() is a Ramanujan theta function.

%I A209939 #25 Feb 16 2025 08:33:16
%S A209939 1,3,0,-2,9,0,-22,0,0,-26,-6,0,25,27,0,46,0,0,26,-66,0,22,0,0,-45,0,0,
%T A209939 0,-78,0,74,-18,0,-122,0,0,-46,75,0,142,81,0,0,0,0,44,138,0,2,0,0,
%U A209939 -194,0,0,-214,78,0,0,-198,0,121,0,0,-146,66,0,52,0,0,22
%N A209939 Expansion of (f(x) * f(x^3))^3 in powers of q where f() is a Ramanujan theta function.
%C A209939 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%C A209939 Number 55 of the 74 eta-quotients listed in Table I of Martin (1996).
%H A209939 Seiichi Manyama, <a href="/A209939/b209939.txt">Table of n, a(n) for n = 0..10000</a>
%H A209939 Y. Martin, <a href="http://dx.doi.org/10.1090/S0002-9947-96-01743-6">Multiplicative eta-quotients</a>, Trans. Amer. Math. Soc. 348 (1996), no. 12, 4825-4856, see page 4852 Table I.
%H A209939 Michael Somos, <a href="/A030203/a030203.txt">Index to Yves Martin's list of 74 multiplicative eta-quotients and their A-numbers</a>
%H A209939 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A209939 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A209939 Expansion of q^(-1/2) * ((eta(q^2) * eta(q^6))^3 / (eta(q) * eta(q^3) * eta(q^4) * eta(q^12)))^3 in powers of q.
%F A209939 Euler transform of period 12 sequence [3, -6, 6, -3, 3, -12, 3, -3, 6, -6, 3, -6, ...].
%F A209939 G.f. is a period 1 Fourier series which satisfies f(-1 / (48 t)) = 48^(3/2) (t/i)^3 f(t) where q = exp(2 Pi i t).
%F A209939 a(n) = b(2*n + 1) where b(n) is multiplicative and b(2^e) = 0^e, b(3^e) = 3^e, b(p^e) = (1 + (-1)^e) / 2 * p^e if p == 5 (mod 6), b(p^e) = b(p) * b(p^(e-1)) - p^2 * b(p^(e-2)) otherwise.
%F A209939 G.f.: ( Product_{k>0} (1 - (-x)^k) * (1 - (-x)^(3*k)) )^3.
%F A209939 a(n) = (-1)^n * A030208(n). a(3*n + 2) = 0. a(3*n + 1) = 3 * a(n).
%F A209939 Conovlution cube of A208978. - _Michael Somos_, Jun 09 2015
%e A209939 G.f. = 1 + 3*x - 2*x^3 + 9*x^4 - 22*x^6 - 26*x^9 - 6*x^10 + 25*x^12 + ...
%e A209939 G.f. = q + 3*q^3 - 2*q^7 + 9*q^9 - 22*q^13 - 26*q^19 - 6*q^21 + 25*q^25 + ...
%t A209939 a[ n_] := SeriesCoefficient[ (QPochhammer[ -x] QPochhammer[ -x^3])^3, {x, 0, n}]; (* _Michael Somos_, Jun 09 2015 *)
%o A209939 (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( ((eta(x^2 + A) * eta(x^6 + A))^3 / (eta(x + A) * eta(x^3 + A) * eta(x^4 + A) * eta(x^12 + A)))^3, n))};
%Y A209939 Cf. A030208, A208978.
%K A209939 sign
%O A209939 0,2
%A A209939 _Michael Somos_, Mar 15 2012