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A247223 Expansion of f(-x^5, -x^7) in powers of x where f() is a Ramanujan theta function.

%I A247223 #35 Jul 13 2025 21:01:18
%S A247223 1,0,0,0,0,-1,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,
%T A247223 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,
%U A247223 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0
%N A247223 Expansion of f(-x^5, -x^7) in powers of x where f() is a Ramanujan theta function.
%H A247223 Seiichi Manyama, <a href="/A247223/b247223.txt">Table of n, a(n) for n = 0..1000</a>
%H A247223 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A247223 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A247223 Euler transform of period 12 sequence [ 0, 0, 0, 0, -1, 0, -1, 0, 0, 0, 0, -1, ...]. - _Michael Somos_, Jan 10 2015
%F A247223 G.f.: Product_{k>0} (1 - x^(12*k)) * (1 - x^(12*k - 5)) * (1 - x^(12*k - 7)).
%e A247223 G.f. = 1 - x^5 - x^7 + x^22 + x^26 - x^51 - x^57 + x^92 + x^100 - x^145 + ...
%e A247223 G.f. = q - q^121 - q^169 + q^529 + q^625 - q^1225 - q^1369 + q^2209 + ...
%t A247223 a[ n_] := SeriesCoefficient[ QPochhammer[q^5,q^12]*QPochhammer[q^7,q^12] *QPochhammer[q^12,q^12], {q, 0, n}]; (* _G. C. Greubel_, Dec 08 2017 *)
%o A247223 (PARI) {a(n) = my(m = 24*n + 1); if( issquare(m, &m) && (m%12==1 || m%12==11), (-1)^((m+6) \ 12))};
%K A247223 sign
%O A247223 0,1
%A A247223 _Michael Somos_, Jan 10 2015