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%I #36 Mar 07 2021 17:38:23
%S 0,1,1,0,0,-1,0,-1,0,0,0,0,1,0,0,1,0,0,0,0,0,0,-1,0,0,0,-1,0,0,0,0,0,
%T 0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,-1,0,0,0,0,0,0,0,
%U 0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0
%N Expansion of 1 - f(-x) in powers of x where f() is a Ramanujan theta function.
%C Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H G. C. Greubel, <a href="/A257628/b257628.txt">Table of n, a(n) for n = 0..10000</a>
%H Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%H Wikipedia, <a href="http://www.wikipedia.org/wiki/Pentagonal number theorem">Pentagonal number theorem</a>
%F G.f.: x + x^2 * (1 - x) + x^3 * (1 - x) * (1 - x^2) + ....
%F G.f.: Sum_{k>0} -(-1)^k * (x^((3*k^2 - k)/2) + x^((3*k^2 + k)/2)).
%F G.f.: Sum_{k>0} -(-1)^k * x^((k^2 + k) / 2) / ((1 - x) * (1 - x^2) * ... * (1 - x^k)).
%F G.f.: -(Product_{j>=1}(1-x^j) - 1), from Euler's Pentagonal Theorem. - _Wolfdieter Lang_, Feb 16 2021
%F a(n) = - A010815(n) unless n=0, a(0) = 0.
%e G.f. = x + x^2 - x^5 - x^7 + x^12 + x^15 - x^22 - x^26 + x^35 + x^40 + ...
%e G.f. = q^25 + q^49 - q^121 - q^169 + q^289 + q^361 - q^529 - q^625 + ...
%t a[ n_] := SeriesCoefficient[ 1 - QPochhammer[ x], {x, 0, n}];
%t a[ n_] := With[ {m = Sqrt[24 n + 1]}, If[ n > 0 && IntegerQ[m], - KroneckerSymbol[ 12, m], 0]];
%o (PARI) {a(n) = if( n<0, 0, polcoeff( 1 - eta(x + x * O(x^n)), n))};
%o (PARI) {a(n) = my(m); if( n>0 && issquare( 24*n + 1, &m), - kronecker( 12, m))};
%Y Cf. A001318, A010815, A341418 (convolution triangle).
%K sign,easy
%O 0
%A _Michael Somos_, Jul 12 2015
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