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A212770 Expansion of q / (chi(q) * chi(q^2) * chi(q^3) * chi(q^6))^2 in powers of q where chi() is a Ramanujan theta function.

%I A212770 #12 Feb 16 2025 08:33:17
%S A212770 1,-2,1,-4,10,-10,12,-24,37,-44,56,-84,126,-160,186,-272,394,-466,568,
%T A212770 -792,1052,-1272,1560,-2040,2663,-3244,3877,-4992,6410,-7644,9180,
%U A212770 -11616,14472,-17284,20712,-25572,31518,-37576,44510,-54416,66402,-78368,92648
%N A212770 Expansion of q / (chi(q) * chi(q^2) * chi(q^3) * chi(q^6))^2 in powers of q where chi() is a Ramanujan theta function.
%C A212770 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H A212770 G. C. Greubel, <a href="/A212770/b212770.txt">Table of n, a(n) for n = 1..2500</a>
%H A212770 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A212770 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A212770 Expansion of b(q) * c(q) * b(q^8) * c(q^8) / (b(q^2) * c(q^2) * b(q^4) * c(q^4))  in powers of q where b(), c() are cubic AGM theta functions.
%F A212770 Expansion of (eta(q) * eta(q^3) * eta(q^8) * eta(q^24) / (eta(q^2) * eta(q^4) * eta(q^6) * eta(q^12)))^2 in powers of q.
%F A212770 Euler transform of period 24 sequence [ -2, 0, -4, 2, -2, 0, -2, 0, -4, 0, -2, 4, -2, 0, -4, 0, -2, 0, -2, 2, -4, 0, -2, 0, ...].
%F A212770 G.f. is a period 1 Fourier series which satisfies f(-1 / (24 t)) = f(t) where q = exp(2 Pi i t).
%F A212770 a(2*n) = -2 * A123647(n). a(4*n) = -4 * A123653(n).
%e A212770 G.f. = x - 2*x^2 + x^3 - 4*x^4 + 10*x^5 - 10*x^6 + 12*x^7 - 24*x^8 + 37*x^9 + ...
%t A212770 a[ n_] := SeriesCoefficient[ q (QPochhammer[ q, -q] QPochhammer[ q^2, -q^2] QPochhammer[ q^3, -q^3] QPochhammer[ q^6, -q^6])^2, {q, 0, n}]; (* _Michael Somos_, Apr 19 2015 *)
%o A212770 (PARI) {a(n) = my(A); if( n<1, 0, n = n-1; A = x * O(x^n); polcoeff( (eta(x + A) * eta(x^3 + A) * eta(x^8 + A) * eta(x^24 + A) / (eta(x^2 + A) * eta(x^4 + A) * eta(x^6 + A) * eta(x^12 + A)))^2, n))};
%Y A212770 Cf. A123647, A123653.
%K A212770 sign
%O A212770 1,2
%A A212770 _Michael Somos_, May 26 2012