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A147702 Expansion of eta(q) * eta(q^10)^3 / (eta(q^2) * eta(q^4) * eta(q^5) * eta(q^20)) in powers of q.

%I A147702 #13 Feb 16 2025 08:33:09
%S A147702 1,-1,0,-1,2,-1,0,-2,4,-3,0,-3,8,-4,0,-6,14,-8,0,-10,22,-12,0,-16,36,
%T A147702 -21,0,-25,56,-30,0,-38,84,-48,0,-57,126,-68,0,-84,184,-102,0,-121,
%U A147702 264,-143,0,-172,376,-207,0,-243,528,-284,0,-338,732,-400,0,-465,1008,-542,0,-636,1374,-744,0,-862,1856,-996,0
%N A147702 Expansion of eta(q) * eta(q^10)^3 / (eta(q^2) * eta(q^4) * eta(q^5) * eta(q^20)) in powers of q.
%C A147702 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H A147702 G. C. Greubel, <a href="/A147702/b147702.txt">Table of n, a(n) for n = 0..1000</a>
%H A147702 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A147702 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A147702 Expansion of chi(-q) * f(q^5) / f(-q^4) in powers of q where f(), chi() are Ramanujan theta functions.
%F A147702 Euler transform of period 20 sequence [ -1, 0, -1, 1, 0, 0, -1, 1, -1, -2, -1, 1, -1, 0, 0, 1, -1, 0, -1, 0, ...].
%F A147702 G.f. is a period 1 Fourier series which satisfies f(-1 / (80 t)) = (5/4)^(-1/2) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A147699.
%F A147702 a(2*n + 1) = - A036026(n). a(4*n) = A138526(n). a(4*n + 2) = 0.
%e A147702 G.f. = 1 - q - q^3 + 2*q^4 - q^5 - 2*q^7 + 4*q^8 - 3*q^9 - 3*q^11 + 8*q^12 + ...
%t A147702 a[ n_] := SeriesCoefficient[ QPochhammer[ -q^5] QPochhammer[ q, q^2] / QPochhammer[ q^4], {q, 0, n}]; (* _Michael Somos_, Sep 02 2015 *)
%o A147702 (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A) * eta(x^10 + A)^3 / (eta(x^2 + A) * eta(x^4 + A) * eta(x^5 + A) * eta(x^20 + A)), n))};
%Y A147702 Cf. A036026, A138526.
%K A147702 sign
%O A147702 0,5
%A A147702 _Michael Somos_, Nov 10 2008