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A227695 Expansion of psi(x)^2 * phi(-x)^6 in powers of x where phi(), psi() are Ramanujan theta functions.

%I A227695 #21 Feb 16 2025 08:33:20
%S A227695 1,-10,37,-50,-30,128,-25,-34,-320,310,410,-370,-87,-410,320,30,500,
%T A227695 384,-630,-640,-359,300,-326,2560,-110,-1098,-1280,-370,1490,-1850,
%U A227695 269,1500,1216,640,570,-3328,340,-2010,-1110,1790,768,3200,303,750,-1600,-442
%N A227695 Expansion of psi(x)^2 * phi(-x)^6 in powers of x where phi(), psi() are Ramanujan theta functions.
%C A227695 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H A227695 G. C. Greubel, <a href="/A227695/b227695.txt">Table of n, a(n) for n = 0..1000</a>
%H A227695 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A227695 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A227695 Expansion of q^(-1/4) * (eta(q)^5 / eta(q^2))^2 in powers of q.
%F A227695 Expansion of phi(-x)^5 * f(-x^2)^3 = phi(-x)^2 * f(-x)^6 in powers of x where phi(), f() are Ramanujan theta functions.
%F A227695 Euler transform of period 2 sequence [ -10, -8, ...].
%F A227695 G.f. is a period 1 Fourier series which satisfies f(-1 / (32 t)) = 8192 (t / i)^4 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A227317.
%F A227695 G.f.: (Product_{k>0} (1 - x^k)^5 / (1 - x^(2*k)))^2.
%F A227695 Convolution of A000729 and A104794.
%e A227695 G.f. = 1 - 10*x + 37*x^2 - 50*x^3 - 30*x^4 + 128*x^5 - 25*x^6 - 34*x^7 - 320*x^8 + ...
%e A227695 G.f. = q - 10*q^5 + 37*q^9 - 50*q^13 - 30*q^17 + 128*q^21 - 25*q^25 - 34*q^29 + ...
%t A227695 a[ n_] := SeriesCoefficient[ (QPochhammer[ x]^5 / QPochhammer[ x^2])^2, {x, 0, n}];
%o A227695 (PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x + A)^5 / eta(x^2 + A))^2, n))};
%Y A227695 Cf. A000729, A104794, A227317.
%K A227695 sign
%O A227695 0,2
%A A227695 _Michael Somos_, Sep 02 2013