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%I #14 Sep 08 2022 08:45:46
%S 1,8,20,0,-70,-64,56,0,-125,160,308,0,110,0,-520,0,57,-560,0,0,182,
%T 512,-880,0,1190,448,884,0,0,0,-1400,0,-1330,-1000,1820,0,-646,-1280,
%U 0,0,-1331,2464,380,0,1120,0,2576,0,0,880,1748,0,-3850,0,-3400,0,2703
%N Expansion of f(q)^8 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 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 Expansion of q^(-1/3) * (eta(q^2)^3 / (eta(q) * eta(q^4)))^8 in powers of q.
%F Euler transform of period 4 sequence [8, -16, 8, -8, ...].
%F G.f. is a period 1 Fourier series which satisfies f(-1 / (36 t)) = 1296 (t/i)^4 f(t) where q = exp(2 Pi i t).
%F a(n) = b(3*n + 1) where b() is multiplicative with b(3^e) = 0^e, b(2^e) = (1+(-1)^e)/2 * -(-8)^(e/2) if e>0, b(p^e) = (1+(-1)^e)/2 * (-p^3)^(e/2) if p == 5 (mod 6), b(p^e) = b(p) * b(p^(e-1)) - b(p^(e-2)) * p^3 if p == 1 (mod 6) where b(p) = (x^2-3*p) * x, 4*p = x^2 + 3 * y^2, |x| < |y| and x == 2 (mod 3).
%F G.f.: Product_{k>0} (1 - (-x)^k)^8.
%F a(n) = (-1)^n * A000731(n).
%F a(4*n + 3) = a(16*n + 13) = 0. a(4*n + 1) = (-1)^n * 8 * a(n).
%F a(2*n) = A153728(n). - _Michael Somos_, Sep 06 2015
%e G.f. = 1 + 8*x + 20*x^2 - 70*x^4 - 64*x^5 + 56*x^6 - 1258*x^8 + ...
%e G.f. = q + 8*q^4 + 20*q^7 - 70*q^13 - 64*q^16 + 56*q^19 - 125*q^25 + 160*q^28 + ...
%t a[ n_] := SeriesCoefficient[ QPochhammer[ -x]^8, {x, 0, n}]; (* _Michael Somos_, Sep 06 2015 *)
%o (PARI) {a(n) = if( n<0, 0, polcoeff( eta(-x + x * O(x^n))^8, n))};
%o (PARI) {a(n) = my(A, p, e, x, y, a0, a1); if( n<0, 0, A = factor(3*n + 1); (-1)^n * prod( k=1, matsize(A)[1], [p, e] = A[k, ]; if( p==3, 0, p%3==2, if( e%2, 0, (-1)^(e/2) * p^(3*e/2)), forstep( y=sqrtint(4*p\3), sqrtint(p\3), -1, if( issquare( 4*p - 3*y^2, &x), if( x%3!=2, x=-x); break)); a0=1; a1 = y = x * (x^2 - 3*p); for( i=2, e, x = y*a1 - p^3*a0; a0=a1; a1=x); a1)))}; /* _Michael Somos_, Sep 06 2015 */
%o (Magma) A := Basis( CuspForms( Gamma0(36), 4), 170); A[1] + 8*A[4] + 20*A[7] - 70*A[12]; /* _Michael Somos_, Sep 02 2015 */
%Y Cf. A000731, A153728.
%K sign
%O 0,2
%A _Michael Somos_, Jun 22 2009
%E Corrected by _Charles R Greathouse IV_, Sep 02 2009
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