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A185125 Expansion of f(-x, x^5) in powers of x where f(,) is the Ramanujan general theta function.

%I A185125 #17 Feb 16 2025 08:33:13
%S A185125 1,-1,0,0,0,1,0,0,-1,0,0,0,0,0,0,0,-1,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,
%T A185125 -1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,-1,
%U A185125 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0
%N A185125 Expansion of f(-x, x^5) in powers of x where f(,) is the Ramanujan general theta function.
%C A185125 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%C A185125 a(n) is nonzero if and only if n is a number of A001082.
%C A185125 The exponents in the q-series for this sequence are the squares of the numbers of A001651.
%H A185125 G. C. Greubel, <a href="/A185125/b185125.txt">Table of n, a(n) for n = 0..1000</a>
%H A185125 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A185125 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A185125 Euler transform of period 24 sequence [ -1, 0, 0, 0, 1, 1, 1, 0, 0, -1, -1, -2, -1, -1, 0, 0, 1, 1, 1, 0, 0, 0, -1, -1, ...].
%F A185125 G.f.: Sum_{k in Z} (-1)^floor(k/2) * x^(k * (3*k + 2)).
%F A185125 a(4*n + 2) = a(4*n + 3) = a(5*n + 2) = a(5*n + 4) = a(8*n + 4) = 0. a(4*n + 1) = - A080902(n). a(8*n) = A010815(n).
%F A185125 a(n) = (-1)^n * A185124(n). - _Michael Somos_, Jun 30 2015
%e A185125 G.f. = 1 - x + x^5 - x^8 - x^16 + x^21 - x^33 + x^40 + x^56 - x^65 + x^85 + ...
%e A185125 G.f. = q - q^4 + q^16 - q^25 - q^49 + q^64 - q^100 + q^121 + q^169 - q^196 + ...
%t A185125 a[ n_] := SeriesCoefficient[ QPochhammer[ x, -x^6] QPochhammer[ -x^5, -x^6] QPochhammer[ -x^6], {x, 0, n}]; (* _Michael Somos_, Jun 30 2015 *)
%o A185125 (PARI) {a(n) = my(m); if( issquare( 3*n + 1, &m), (m%3!=0) * (-1)^((m+3) \ 6 + n), 0)};
%Y A185125 Cf. A010815, A080902, A185124.
%K A185125 sign
%O A185125 0,1
%A A185125 _Michael Somos_, Jan 20 2012