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A128591 Expansion of  f(x, x^5) * f(x, x^3) in powers of x where f(, ) is Ramanujan's general theta function. 7

%I

%S 1,2,1,1,1,1,2,1,2,1,1,3,0,0,1,2,2,1,1,1,1,2,3,1,1,0,2,1,1,2,0,2,0,2,

%T 1,0,4,2,0,1,1,2,1,2,2,1,2,0,1,1,2,0,1,1,1,2,2,2,0,1,1,3,1,1,0,1,4,1,

%U 2,1,0,4,0,0,1,1,2,1,2,1,1,0,1,1,1,2,3

%N Expansion of f(x, x^5) * f(x, x^3) in powers of x where f(, ) is Ramanujan's general 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="/A128591/b128591.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 chi(x) * psi(x) * psi(-x^3) in powers of x where psi(), chi() are Ramanujan theta functions. - _Michael Somos_, Nov 15 2015

%F Expansion of q^(-11/24) * eta(q^2)^4 * eta(q^3) * eta(q^12) / (eta(q)^2 * eta(q^4) * eta(q^6)) in powers of q.

%F Euler transform of period 12 sequence [ 2, -2, 1, -1, 2, -2, 2, -1, 1, -2, 2, -2, ...].

%F a(n) = A128582(4*n + 1).

%F 2 * a(n) = A257920(3*n + 1). - a(n) = A260118(4*n + 1). 2 * a(n) = A257921(6*n + 2). -2 * a(n) = A128580(12*n + 5) = A190615(12*n + 5). - _Michael Somos_, Nov 15 2015

%e G.f. = 1 + 2*x + x^2 + x^3 + x^4 + x^5 + 2*x^6 + x^7 + 2*x^8 + x^9 + x^10 + 3*x^11 + ...

%e G.f. = q^11 + 2*q^35 + q^59 + q^83 + q^107 + q^131 + 2*q^155 + q^179 + 2*q^203 + ...

%t a[ n_] := SeriesCoefficient[ 2^(-3/2) x^(-1/2) QPochhammer[ -x, x^2] EllipticTheta[ 2, 0, x^(1/2)] EllipticTheta[ 2, Pi/4, x^(3/2)], {x, 0, n}]; (* _Michael Somos_, Nov 15 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) / (eta(x + A)^2 * eta(x^4 + A) * eta(x^6 + A)), n))};

%Y Cf. A128580, A128582, A190615, A257920, A257921,A260118.

%K nonn

%O 0,2

%A _Michael Somos_, Mar 11 2007

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Last modified August 5 14:42 EDT 2021. Contains 346469 sequences. (Running on oeis4.)