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A216711 Expansion of q * (phi(q) * psi(-q))^8 in powers of q where phi(), psi() are Ramanujan theta functions.

%I A216711 #25 Feb 16 2025 08:33:18
%S A216711 1,8,12,-64,-210,96,1016,512,-2043,-1680,1092,-768,1382,8128,-2520,
%T A216711 -4096,14706,-16344,-39940,13440,12192,8736,68712,6144,-34025,11056,
%U A216711 -50760,-65024,-102570,-20160,227552,32768,13104,117648,-213360,130752,160526,-319520
%N A216711 Expansion of q * (phi(q) * psi(-q))^8 in powers of q where phi(), psi() are Ramanujan theta functions.
%C A216711 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H A216711 Seiichi Manyama, <a href="/A216711/b216711.txt">Table of n, a(n) for n = 1..10000</a>
%H A216711 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A216711 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A216711 Expansion of (eta(q^2) / (eta(q) * eta(q^4)))^8 in powers of q.
%F A216711 a(n) is multiplicative with a(2) = 8, a(2^e) = -(-8)^e if e>1, a(p^e) = a(p) * a(p^(e-1)) - p^7 * a(p^(e-2)) if p>2.
%F A216711 G.f. is a period 1 Fourier series which satisfies f(-1 / (4 t)) = 256 (t/i)^8 f(t) where q = exp(2 Pi i t).
%F A216711 a(n) = -(-1)^n * A002288(n). Convolution square of A134461.
%e A216711 G.f. = q + 8*q^2 + 12*q^3 - 64*q^4 - 210*q^5 + 96*q^6 + 1016*q^7 + 512*q^8 + ...
%t A216711 a[ n_] := SeriesCoefficient[ (EllipticTheta[ 4, 0, q^2] EllipticTheta[ 2, 0, q^(1/2)] / 2)^8, {q, 0, n}];
%o A216711 (PARI) {a(n) = my(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( ( eta(x^2 + A)^4 / eta(x + A) / eta(x^4 + A) )^8, n))};
%o A216711 (Magma) A := Basis( CuspForms( Gamma0(4), 8), 39); A[1] + 8*A[2]; /* _Michael Somos_, Jun 10 2015 */
%Y A216711 Cf. A002288, A134461.
%K A216711 sign,mult,look
%O A216711 1,2
%A A216711 _Michael Somos_, Apr 10 2013