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A208451 Expansion of phi(q) * phi(-q)^3 in powers of q where phi() is a Ramanujan theta function.

%I A208451 #19 Feb 16 2025 08:33:16
%S A208451 1,-4,0,16,-8,-24,0,32,24,-52,0,48,-32,-56,0,96,24,-72,0,80,-48,-128,
%T A208451 0,96,96,-124,0,160,-64,-120,0,128,24,-192,0,192,-104,-152,0,224,144,
%U A208451 -168,0,176,-96,-312,0,192,96,-228,0,288,-112,-216,0,288,192,-320
%N A208451 Expansion of phi(q) * phi(-q)^3 in powers of q where phi() is a Ramanujan theta function.
%C A208451 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H A208451 G. C. Greubel, <a href="/A208451/b208451.txt">Table of n, a(n) for n = 0..1000</a>
%H A208451 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A208451 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A208451 Expansion of phi(-q^4)^4 - 4 * q * psi(-q^2)^4 = phi(q) * phi(-q)^3 = phi(-q)^2 * phi(-q^2)^2 = phi(-q^2)^6 / phi(q)^2 = psi(-q)^4 * chi(-q^2)^6 = f(-q)^4 * chi(-q^2)^2 = f(-q)^6 / psi(-q)^2 in powers of q where phi(), psi(), chi(), f() are Ramanujan theta functions.
%F A208451 Expansion of (eta(q)^2 * eta(x^2) / eta(x^4))^2 in powers of q.
%F A208451 Euler transform of period 4 sequence [ -4, -6, -4, -4, ...].
%F A208451 G.f. is a period 1 Fourier series which satisfies f(-1 / (16 t)) = 5128 (t/i)^2 g(t) where q = exp(2 Pi i t) and g(t) is the g.f. for A097723.
%F A208451 a(4*n + 2) = 0. a(2*n + 1) = -4 * A121613(n). a(4*n) = A096727(n). a(4*n + 1) = -4 * A112610(n). a(4*n + 3) = 16 * A097723(n). a(8*n) = A004011(n). a(8*n + 4) = -8 * A008438(n).
%e A208451 G.f. = 1 - 4*q + 16*q^3 - 8*q^4 - 24*q^5 + 32*q^7 + 24*q^8 - 52*q^9 + 48*q^11 + ...
%t A208451 a[ n_] := SeriesCoefficient[ (QPochhammer[ q]^2 QPochhammer[ q^2] / QPochhammer[ q^4])^2, {q, 0, n}]; (* _Michael Somos_, Aug 21 2014 *)
%o A208451 (PARI) {a(n) = if( n<1, n==0, if( n%4 == 2, 0, -4 * if( n%2, (-1)^(n\2) * sigma(n), -2 * (-1)^(n\4) * sumdiv( n\4, d, if( d%4, d)))))};
%o A208451 (PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x + A)^2 * eta(x^2 + A) / eta(x^4 + A))^2, n))};
%o A208451 (Magma) A := Basis( ModularForms( Gamma0(16), 2), 58); A[1] - 4*A[2] + 16*A[4] - 8*A[5]; /* _Michael Somos_, Aug 21 2014 */
%Y A208451 Cf. A004011, A008438, A096727, A097723, A112610, A121613.
%K A208451 sign
%O A208451 0,2
%A A208451 _Michael Somos_, Feb 26 2012