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A134178 Expansion of chi(x) * chi(-x^2)^2 * chi(-x^3) * chi(-x^4) * chi(x^6)^2 * chi(-x^12) in powers of x where chi() is a Ramanujan theta function. 3

%I #13 Mar 12 2021 22:24:45

%S 1,1,-2,-2,0,1,2,0,0,-1,-4,0,1,0,6,2,0,1,-8,0,0,0,12,0,-1,-1,-18,-4,0,

%T -1,24,0,0,2,-32,0,0,1,44,6,0,-2,-58,0,0,-1,76,0,1,2,-100,-8,0,1,128,

%U 0,0,-3,-164,0,0,-1,210,12,0,4,-264,0,0,2,332,0,-1,-5,-416,-18,0,-2,516,0,0,5,-640,0,-1,2,790,24

%N Expansion of chi(x) * chi(-x^2)^2 * chi(-x^3) * chi(-x^4) * chi(x^6)^2 * chi(-x^12) in powers of x where chi() is a Ramanujan theta function.

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

%H G. C. Greubel, <a href="/A134178/b134178.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 Euler transform of period 24 sequence [ 1, -3, 0, -1, 1, -1, 1, 0, 0, -3, 1, -4, 1, -3, 0, 0, 1, -1, 1, -1, 0, -3, 1, 0, ...].

%F a(12*n + 4) = a(12*n + 7) = a(12*n + 8) = a(12*n + 11) = 0.

%F a(4*n + 1) = a(12*n) = A029838(n). a(4*n + 2) = a(12*n + 3) = -2 * A083365(n).

%e G.f. = 1 + x - 2*x^2 - 2*x^3 + x^5 + 2*x^6 - x^9 - 4*x^10 + x^12 + 6*x^14 + ...

%e G.f. = q^-3 + q^-1 - 2*q - 2*q^3 + q^7 + 2*q^9 - q^15 - 4*q^17 + q^21 + 6*q^25 + ...

%t a[ n_] := SeriesCoefficient[ QPochhammer[ -x, x^2] QPochhammer[ x^2, x^4]^2 QPochhammer[ x^3, x^6] QPochhammer[ x^4, x^8] QPochhammer[-x^6, x^12]^2 QPochhammer[ x^12, x^24], {x, 0, n}]; (* _Michael Somos_, Oct 25 2015 *)

%t a[ n_] := SeriesCoefficient[ QPochhammer[ -x^12, x^24] QPochhammer[ x^24, x^48] + x QPochhammer[ -x^4, x^8] QPochhammer[ x^8, x^16] - 2 x^2 QPochhammer[ x^16] / QPochhammer[ -x^4] - 2 x^3 QPochhammer[ x^48] / QPochhammer[ -x^12], {x, 0, n}]; (* _Michael Somos_, Oct 25 2015 *)

%o (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^4 * eta(x^3 + A) * eta(x^12 + A)^5 / (eta(x + A) * eta(x^4 + A)^2 * eta(x^6 + A)^3 * eta(x^8 + A) * eta(x^24 + A)^3), n))};

%Y Cf. A029838, A083365.

%K sign

%O 0,3

%A _Michael Somos_, Oct 11 2007

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