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A227317 Expansion of psi(x)^6 * phi(-x)^2 in powers of x where phi(), psi() are Ramanujan theta functions.

%I A227317 #21 Feb 16 2025 08:33:20
%S A227317 1,2,-5,-10,5,6,10,40,-20,-50,19,-52,-30,50,-25,74,97,50,-25,-140,69,
%T A227317 -34,-100,-50,-185,-6,83,310,-60,-60,410,-128,145,-100,-245,250,-87,
%U A227317 -90,-400,-410,-151,362,185,-50,285,30,150,-240,500,370,-68,222,5,-190
%N A227317 Expansion of psi(x)^6 * phi(-x)^2 in powers of x where phi(), psi() are Ramanujan theta functions.
%C A227317 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H A227317 G. C. Greubel, <a href="/A227317/b227317.txt">Table of n, a(n) for n = 0..1000</a>
%H A227317 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A227317 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A227317 Expansion of psi(x)^5 * f(-x)^3 = psi(x)^2 * f(-x^2)^6 in powers of x where psi(), f() are Ramanujan theta functions.
%F A227317 Expansion of q^(-3/4) * (eta(q^2)^5 / eta(x))^2 in powers of q.
%F A227317 Euler transform of period 2 sequence [ 2, -8, ...].
%F A227317 G.f. is a period 1 Fourier series which satisfies f(-1 / (32 t)) = 128 (t / i)^4 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A227695.
%F A227317 G.f.: (Product_{k>0} (1 - x^(2*k))^5 / (1 - x^k))^2.
%F A227317 Convolution of A008439 and A010816.
%F A227317 -8 * a(n) = A215600(2*n + 1).
%e A227317 1 + 2*x - 5*x^2 - 10*x^3 + 5*x^4 + 6*x^5 + 10*x^6 + 40*x^7 - 20*x^8 + ...
%e A227317 q^3 + 2*q^7 - 5*q^11 - 10*q^15 + 5*q^19 + 6*q^23 + 10*q^27 + 40*q^31 + ...
%t A227317 a[ n_] := SeriesCoefficient[ (QPochhammer[ q^2]^5 / QPochhammer[ q])^2, {q, 0, n}]
%o A227317 (PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x^2 + A)^5 / eta(x + A))^2, n))}
%Y A227317 Cf. A008439, A010816, A215600, A227695.
%K A227317 sign
%O A227317 0,2
%A A227317 _Michael Somos_, Sep 02 2013