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A263074 Expansion of phi(-x) / (chi(-x^3) * chi(-x^5)) in powers of x where phi(), chi() are Ramanujan theta functions.

%I A263074 #10 Feb 16 2025 08:33:27
%S A263074 1,-2,0,1,0,1,-1,0,1,0,-1,-1,0,1,0,1,-3,0,2,0,1,-1,0,2,0,0,-3,0,1,0,2,
%T A263074 -4,0,2,0,1,-3,0,3,0,1,-4,0,2,0,3,-6,0,4,0,4,-6,0,4,0,1,-7,0,4,0,3,-9,
%U A263074 0,5,0,4,-8,0,6,0,3,-10,0,6,0,6,-13,0,8,0,5
%N A263074 Expansion of phi(-x) / (chi(-x^3) * chi(-x^5)) in powers of x where phi(), chi() are Ramanujan theta functions.
%C A263074 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H A263074 G. C. Greubel, <a href="/A263074/b263074.txt">Table of n, a(n) for n = 0..1000</a>
%H A263074 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A263074 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A263074 Expansion of q^(-1/3) * eta(q)^2 * eta(q^6) * eta(q^10) / (eta(q^2) * eta(q^3) * eta(q^5)) in powers of q.
%F A263074 Euler transform of period 30 sequence [ -2, -1, -1, -1, -1, -1, -2, -1, -1, -1, -2, -1, -2, -1, 0, -1, -2, -1, -2, -1, -1, -1, -2, -1, -1, -1, -1, -1, -2, -1, ...].
%F A263074 G.f.: Product_{k>0} (1 + x^(3*k)) * (1 + x^(5*k)) * (1 - x^k) / (1 + x^k).
%F A263074 Convolution inverse is A100823.
%F A263074 a(5*n + 2) = a(5*n + 4) = 0. a(5*n + 3) = A263073(n).
%e A263074 G.f. = 1 - 2*x + x^3 + x^5 - x^6 + x^8 - x^10 - x^11 + x^13 + x^15 + ...
%e A263074 G.f. = q - 2*q^4 + q^10 + q^16 - q^19 + q^25 - q^31 - q^34 + q^40 + ...
%t A263074 a[ n_] := SeriesCoefficient[ EllipticTheta[ 4, 0, x] / (QPochhammer[ x^3, x^6] QPochhammer[ x^5, x^10]), {x, 0, n}];
%o A263074 (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A)^2 * eta(x^6 + A) * eta(x^10 + A) / (eta(x^2 + A) * eta(x^3 + A) * eta(x^5 + A)), n))};
%Y A263074 Cf. A100823, A263073.
%K A263074 sign
%O A263074 0,2
%A A263074 _Michael Somos_, Oct 08 2015