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A244525 Expansion of f(-x^1, -x^7) in powers of x where f(, ) is Ramanujan's general theta function.

%I A244525 #26 Feb 16 2025 08:33:23
%S A244525 1,-1,0,0,0,0,0,-1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,-1,0,0,0,0,0,
%T A244525 0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,
%U A244525 0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0
%N A244525 Expansion of f(-x^1, -x^7) in powers of x where f(, ) is Ramanujan's general theta function.
%H A244525 Seiichi Manyama, <a href="/A244525/b244525.txt">Table of n, a(n) for n = 0..10000</a>
%H A244525 C. G. J. Jacobi, <a href="https://archive.org/stream/gesammelwerke01jacorich#page/n363/mode/2up">Uber die Zur Numerischen Berechnung Der Elliptischen Funtionen Zweckmassigsten Formeln</a>, in Gesammelte Werke, Bd. I, 1881, pp. 343-368. See p. 347 equ. (7.)
%H A244525 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A244525 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A244525 Euler transform of period 8 sequence [-1, 0, 0, 0, 0, 0, -1, -1, ...].
%F A244525 G.f.: f(-x, -x^7) = Sum_{k in Z} (-1)^k * x^(4*k^2 - 3*k).
%F A244525 a(n) = (-1)^n * A214263(n).
%F A244525 G.f.: Product_{k>0} (1 - x^(8*k-1)) * (1 - x^(8*k-7)) * (1 - x^(8*k)). - _Seiichi Manyama_, Jun 14 2016
%e A244525 G.f. = 1 - x - x^7 + x^10 + x^22 - x^27 - x^45 + x^52 + x^76 - x^85 + ...
%e A244525 G.f. = q^9 - q^25 - q^121 + q^169 + q^361 - q^441 - q^729 + q^841 + ...
%t A244525 a[ n_] := SeriesCoefficient[ QPochhammer[ x^1, x^8] QPochhammer[ x^7, x^8] QPochhammer[ x^8], {x, 0, n}];
%o A244525 (PARI) {a(n) = issquare(16*n + 9) * (-1)^n};
%Y A244525 Cf. A214263.
%K A244525 sign
%O A244525 0,1
%A A244525 _Michael Somos_, Jun 29 2014