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A260168 Expansion of f(-x)^6 * f(-x^3)^2 / phi(-x^3)^8 in powers of q where phi(), f() are Ramanujan theta functions. 1

%I

%S 1,-6,9,24,-114,126,262,-1044,999,1852,-6672,5868,10103,-34134,28341,

%T 46336,-149400,118872,186926,-581412,447507,682340,-2062332,1545336,

%U 2297737,-6782508,4970241,7236280,-20938728,15056694,21531158,-61246128,43329078,61003980

%N Expansion of f(-x)^6 * f(-x^3)^2 / phi(-x^3)^8 in powers of q 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="/A260168/b260168.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^(-1/2) * (eta(q)^3 * eta(q^6)^4 / eta(q^3)^7)^2 in powers of q.

%F Euler transform of period 6 sequence [ -6, -6, 8, -6, -6, 0, ...].

%F -2 * a(n) = A261576(2*n + 1).

%e G.f. = 1 - 6*x + 9*x^2 + 24*x^3 - 114*x^4 + 126*x^5 + 262*x^6 - 1044*x^7 + ...

%e G.f. = q - 6*q^3 + 9*q^5 + 24*q^7 - 114*q^9 + 126*q^11 + 262*q^13 - 1044*q^15 + ...

%t a[ n_] := SeriesCoefficient[ QPochhammer[ x]^6 QPochhammer[ x^3]^2 / EllipticTheta[ 4, 0, x^3]^8, {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^6 + A)^4 / eta(x^3 + A)^7)^2, n))};

%Y Cf. A261576.

%K sign

%O 0,2

%A _Michael Somos_, Nov 09 2015

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Last modified June 16 21:55 EDT 2021. Contains 345080 sequences. (Running on oeis4.)