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A258940 Expansion of f(-x^8) * f(-x^12) * f(-x^24) * f(-x^2, -x^6)^2 / (f(-x^2) * f(-x^3, -x^5) * f(-x^3, -x^21)) in powers of x where f() is a Ramanujan theta function and f(, ) is Ramanujan's general theta function.

%I A258940 #15 Feb 16 2025 08:33:25
%S A258940 1,0,-1,2,1,-1,1,1,0,1,0,0,2,0,-1,2,1,-2,1,1,0,1,-1,0,3,0,-1,2,1,-1,1,
%T A258940 2,0,1,0,0,2,-1,-2,2,1,-1,0,1,0,2,0,0,2,-1,-1,2,2,-1,1,1,0,0,1,0,2,0,
%U A258940 -2,2,1,-1,2,1,0,1,0,0,2,0,-1,2,0,-2,1,1,0
%N A258940 Expansion of f(-x^8) * f(-x^12) * f(-x^24) * f(-x^2, -x^6)^2 / (f(-x^2) * f(-x^3, -x^5) * f(-x^3, -x^21)) in powers of x where f() is a Ramanujan theta function and f(, ) is Ramanujan's general theta function.
%C A258940 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H A258940 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A258940 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A258940 Euler transform of period 24 sequence [ 0, -1, 2, 1, 1, -1, 0, -1, 0, -1, 1, 0, 1, -1, 0, -1, 0, -1, 1, 1, 2, -1, 0, -2, ...].
%F A258940 a(3*n + 2) = - A128582(n).
%F A258940 a(12*n + 8) = a(12*n + 11) = 0.
%e A258940 G.f. = 1 - x^2 + 2*x^3 + x^4 - x^5 + x^6 + x^7 + x^9 + 2*x^12 - x^14 + ...
%e A258940 G.f. = q - q^5 + 2*q^7 + q^9 - q^11 + q^13 + q^15 + q^19 + 2*q^25 - q^29 + ...
%t A258940 a[ n_] := SeriesCoefficient[ Product[(1 - x^k)^-{ 0, -1, 2, 1, 1, -1, 0, -1, 0, -1, 1, 0, 1, -1, 0, -1, 0, -1, 1, 1, 2, -1, 0, -2}[[Mod[k, 24, 1]]], {k, n}], {x, 0, n}];
%o A258940 (PARI) {a(n) = if( n<0, 0, polcoeff( prod(k=1, n, (1 - x^k + x * O(x^n))^[ 2, 0, 1, -2, -1, -1, 1, 0, 1, 0, 1, -1, 0, -1, 1, 0, 1, 0, 1, -1, -1, -2, 1, 0][k%24 + 1]), n))};
%Y A258940 Cf. A128582.
%K A258940 sign
%O A258940 0,4
%A A258940 _Michael Somos_, Nov 07 2015