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A209676 Expansion of f(x)^12 in powers of x where f() is a Ramanujan theta function.

%I A209676 #29 Feb 16 2025 08:33:16
%S A209676 1,12,54,88,-99,-540,-418,648,594,-836,1056,4104,-209,-4104,-594,
%T A209676 -4256,-6480,4752,-298,-5016,17226,12100,-5346,1296,-9063,7128,19494,
%U A209676 -29160,-10032,7668,-34738,-8712,-22572,-21812,49248,46872,67562,-2508,-47520,76912
%N A209676 Expansion of f(x)^12 in powers of x where f() is a Ramanujan theta function.
%C A209676 Number 35 of the 74 eta-quotients listed in Table I of Martin (1996). See g.f. B(q) below: cusp form weight 6 level 16.
%C A209676 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H A209676 G. C. Greubel, <a href="/A209676/b209676.txt">Table of n, a(n) for n = 0..2500</a>
%H A209676 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 A209676 Michael Somos, <a href="/A030203/a030203.txt">Index to Yves Martin's list of 74 multiplicative eta-quotients and their A-numbers</a>
%H A209676 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A209676 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A209676 Expansion of q^(-1/2) * (eta(q^2)^3 / (eta(q) * eta(q^4)))^12 in powers of q.
%F A209676 Euler transform of period 4 sequence [ 12, -24, 12, -12, ...].
%F A209676 a(n) = b(2*n + 1) where b(n) is multiplicative and b(2^e) = 0^e, b(p^e) = b(p) * b(p^(e-1)) - p^5 * b(p^(e-2)) otherwise.
%F A209676 G.f. is a period 1 Fourier series which satisfies f(-1 / (16 t)) = 4096 (t/i)^6 f(t) where q = exp(2 Pi i t).
%F A209676 G.f.: (Product_{k>0} (1 - (-x)^k))^12.
%F A209676 a(n) = (-1)^n * A000735(n).
%F A209676 Convolution cube of A187076. Convolution fourth power of A133089. Convolution twelfth power of A121373.
%e A209676 G.f. = 1 + 12*x + 54*x^2 + 88*x^3 - 99*x^4 - 540*x^5 - 418*x^6 + 648*x^7 + ...
%e A209676 G.f. B(q) of {b(n)}: q + 12*q^3 + 54*q^5 + 88*q^7 - 99*q^9 - 540*q^11 - 418*q^13 + ...
%t A209676 a[ n_] := SeriesCoefficient[ QPochhammer[ -x]^12, {x, 0, n}]; (* _Michael Somos_, Jun 09 2015 *)
%o A209676 (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x^2 + A)^3 / (eta(x + A) * eta(x^4 + A)))^12, n))};
%o A209676 (Magma) A := Basis( CuspForms( Gamma0(16), 6), 81); A[1] + 12*A[3] + 54*A[5] + 88*A[7]; /* _Michael Somos_, Jun 09 2015 */
%Y A209676 Cf. A121373, A133089, A187076.
%Y A209676 A000735 is the same except for signs.
%K A209676 sign
%O A209676 0,2
%A A209676 _Michael Somos_, Mar 11 2012