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%I #14 Sep 08 2022 08:46:06
%S 1,-1,-2,1,0,2,2,-1,-2,-1,2,-1,-2,0,0,0,1,3,2,-2,2,-6,-4,3,0,4,0,3,2,
%T 0,-4,0,-2,-2,0,-3,0,2,0,0,4,-5,-2,1,6,0,4,0,-3,2,2,5,-8,2,4,-6,0,3,
%U -4,-9,-8,0,8,0,-2,-5,4,6,0,10,-2,4,6,-3,-6,2,-2
%N Expansion of q^2 * f(-q) * f(-q^4) * f(-q^16) * f(-q^2, -q^14) in powers of q 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="/A229893/b229893.txt">Table of n, a(n) for n = 2..2500</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>
%F Euler transform of period 16 sequence [-1, -2, -1, -2, -1, -1, -1, -2, -1, -1, -1, -2, -1, -2, -1, -4, ...].
%F a(2^n) = A108520(n-1). a(16*n + 1) = a(16*n + 15) = 0. -2 * a(n) = A229502(2*n). a(8*n) = 2 * A229502(n).
%e G.f. = q^2 - q^3 - 2*q^4 + q^5 + 2*q^7 + 2*q^8 - q^9 - 2*q^10 - q^11 + ...
%t a[ n_] := SeriesCoefficient[ q^2 QPochhammer[ q^2, q^16] QPochhammer[ q^14, q^16] QPochhammer[ q^16]^2 QPochhammer[ q] QPochhammer[ q^4], {q, 0, n}]
%o (PARI) {a(n) = local(A, m); if( n<2, 0, n-=2; A = x * O(x^n); polcoeff( eta(x + A) * eta(x^4 + A) * eta(x^16 + A) * sum( k=0, n\2, if( issquare( 16*k + 9, &m), (-1)^k * x^(2*k), 0), A), n))}
%o (Sage) CuspForms( Gamma1(16), 2, prec=79).1
%o (Magma) Basis( CuspForms( Gamma1(16), 2), 79)[2];
%Y Cf. A108520, A229502.
%K sign
%O 2,3
%A _Michael Somos_, Oct 02 2013
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