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%I A127786 #19 Mar 12 2021 22:24:44
%S A127786 1,2,2,4,0,-4,0,-8,-2,6,-8,4,0,-12,0,-8,-4,8,10,12,0,-8,0,-8,8,14,-8,
%T A127786 16,0,-4,0,-16,6,16,16,8,0,-20,0,-8,-8,8,-16,20,0,-20,0,-16,-8,18,10,
%U A127786 8,0,-12,0,-24,0,16,-24,12,0,-20,0,-24,12,8,16,28,0,-16,0,-8,-10,32,-8,20,0,-16,0,-16,-8,18,32,20,0,-24,0
%N A127786 Expansion of phi(q) * phi(q^2) * phi(-q^4) in powers of q where phi() is a Ramanujan theta function.
%C A127786 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H A127786 G. C. Greubel, <a href="/A127786/b127786.txt">Table of n, a(n) for n = 0..5000</a>
%H A127786 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A127786 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A127786 Expansion of eta(q^2)^3 * eta(q^4)^5 / (eta(q)^2 * eta(q^8)^3) in powers of q.
%F A127786 Euler transform of period 8 sequence [ 2, -1, 2, -6, 2, -1, 2, -3, ...].
%F A127786 G.f. is a period 1 Fourier series which satisfies f(-1 / (64 t)) = 128 * (t/i)^(3/2) g(t) where q = exp(2 Pi i t) and g() is the g.f. of A213622. - _Michael Somos_, Sep 08 2014
%F A127786 a(8*n + 4) = a(8*n + 6) = 0.
%F A127786 a(n) = A080963(2*n). a(2*n) = A116597(n). a(2*n + 1) = 2 * A246836(n). a(4*n + 1) = 2 * A246835(n). a(4*n + 3) = 4 * A246833(n). - _Michael Somos_, Sep 08 2014
%F A127786 a(8*n) = A212885(n). a(8*n + 1) = 2 * A213622(n). a(8*n + 2) = 2 * A246954(n). a(8*n + 3) = 4 * A246832(n). a(8*n + 5) = - 4 * A246837(n). a(8*n + 7) = - 8 * A033763(n). - _Michael Somos_, Sep 08 2014
%F A127786 a(3*n + 2) = 2 * A257873(n). - _Michael Somos_, May 11 2015
%e A127786 G.f. = 1 + 2*q + 2*q^2 + 4*q^3 - 4*q^5 - 8*q^7 - 2*q^8 + 6*q^9 - 8*q^10 + ...
%t A127786 a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, q] EllipticTheta[ 3, 0, q^2] EllipticTheta[ 4, 0, q^4], {q, 0, n}]; (* _Michael Somos_, Sep 08 2014 *)
%o A127786 (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^3 * eta(x^4 + A)^5 / (eta(x + A)^2 * eta(x^8 + A)^3), n))};
%Y A127786 Cf. A033763, A080963, A116597, A212885, A213622, A246832, A246833, A246835, A246836, A246837, A246954, A257873.
%K A127786 sign
%O A127786 0,2
%A A127786 _Michael Somos_, Jan 29 2007

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