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A258771 Expansion of psi(-x) * phi(x)^4 in powers of x where phi(), psi() are Ramanujan theta functions.

%I A258771 #11 Feb 16 2025 08:33:25
%S A258771 1,7,16,7,-16,0,17,-48,-64,16,1,-16,16,-32,32,55,-48,64,64,16,128,-9,
%T A258771 -80,-32,16,48,-80,96,49,-144,-16,-144,-64,-64,-96,144,33,-64,-160,0,
%U A258771 112,32,32,-96,128,-25,0,32,-160,304,144,96,144,-48,48,119,16,-256
%N A258771 Expansion of psi(-x) * phi(x)^4 in powers of x where phi(), psi() are Ramanujan theta functions.
%C A258771 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H A258771 G. C. Greubel, <a href="/A258771/b258771.txt">Table of n, a(n) for n = 0..1000</a>
%H A258771 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A258771 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A258771 Expansion of q^(-1/8) * eta(q^2)^19 / (eta(q) * eta(q^4))^7 in powers of q.
%F A258771 Euler transform of period 4 sequence [ 7, -12, 7, -5, ...].
%F A258771 G.f. is a period 1 Fourier series which satisfies f(-1 / (256 t)) = 1024 (t/i)^(5/2) f(t) where q = exp(2 Pi i t).
%F A258771 a(3*n + 2) = 16 * A258770(n).
%F A258771 Convolution square is A209942.
%e A258771 G.f. = 1 + 7*x + 16*x^2 + 7*x^3 - 16*x^4 + 17*x^6 - 48*x^7 - 64*x^8 + ...
%e A258771 G.f. = q + 7*q^9 + 16*q^17 + 7*q^25 - 16*q^33 + 17*q^49 - 48*q^57 + ...
%t A258771 a[ n_] := SeriesCoefficient[EllipticTheta[ 3, 0, x]^4 QPochhammer[ x] / QPochhammer[ x^2, x^4], {x, 0, n}];
%o A258771 (PARI) {a(n) = my(A);  if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^19 / (eta(x + A) * eta(x^4 + A) )^7, n))};
%Y A258771 Cf. A209942, A258770.
%K A258771 sign
%O A258771 0,2
%A A258771 _Michael Somos_, Jun 09 2015