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 A259790 Expansion of f(-x)^3 * phi(x^2) in powers of x where phi(), f() are Ramanujan theta functions. 1

%I

%S 1,-3,2,-1,0,10,-7,0,-12,-6,9,10,18,0,-14,-11,0,-22,20,-6,0,23,0,4,

%T -14,0,0,0,3,26,-30,0,2,-28,0,10,-13,0,20,26,0,0,18,0,34,-19,-30,-60,

%U 0,0,2,-6,0,-2,34,21,-14,42,0,0,-12,0,0,4,0,-22,-23,0,-12

%N Expansion of f(-x)^3 * phi(x^2) in powers of x where phi(), f() are Ramanujan theta functions.

%C Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

%H G. C. Greubel, <a href="/A259790/b259790.txt">Table of n, a(n) for n = 0..1000</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 Expansion of q^(-3/8) * eta(q)^3 * eta(q^4)^5 / (eta(q^2)^2 * eta(q^8)^2) in powers of q.

%F Euler transform of period 8 sequence [ -3, -1, -3, -6, -3, -1, -3, -4, ...].

%F G.f.: Product_{k>0} (1 - x^k)^3 * (1 + x^(2*k))^2 * (1 - x^(4*k)) / (1 + x^(4*k))^2.

%e G.f. = 1 - 3*x + 2*x^2 - x^3 + 10*x^5 - 7*x^6 - 12*x^8 - 6*x^9 + 9*x^10 + ...

%e G.f. = q^3 - 3*q^11 + 2*q^19 - q^27 + 10*q^43 - 7*q^51 - 12*q^67 - 6*q^75 + ...

%t a[ n_] := SeriesCoefficient[ QPochhammer[ x]^3 EllipticTheta[ 3, 0, x^2], {x, 0, n}];

%o (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A)^3 * eta(x^4 + A)^5 / (eta(x^2 + A)^2 * eta(x^8 + A)^2), n))};

%K sign

%O 0,2

%A _Michael Somos_, Jul 05 2015

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Last modified April 15 12:02 EDT 2021. Contains 342977 sequences. (Running on oeis4.)