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A266684 Expansion of f(-x) * f(-x^2)^4 / psi(x^3) in powers of x where psi(), f() are Ramanujan theta functions. 1

%I #9 Mar 12 2021 22:24:48

%S 1,-1,-5,3,7,4,3,-18,-17,-1,20,36,-9,-14,-18,-12,31,16,-5,-54,-28,6,

%T 36,72,15,-21,-70,3,54,28,-12,-90,-65,-12,80,72,7,-38,-54,42,68,40,30,

%U -126,-108,4,72,144,-33,-43,-105,-48,98,52,3,-144,-90,18,140,180

%N Expansion of f(-x) * f(-x^2)^4 / psi(x^3) in powers of x where psi(), f() are Ramanujan theta functions.

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

%H Alois P. Heinz, <a href="/A266684/b266684.txt">Table of n, a(n) for n = 0..10000</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 eta(q) * eta(q^2)^4 * eta(q^3) / eta(q^6)^2 in powers of q.

%F Euler transform of period 6 sequence [ -1, -5, -2, -5, -1, -4, ...].

%F G.f. is a period 1 Fourier series which satisfies f(-1 / (24 t)) = 248832^(1/2) (t/I)^2 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A263021.

%F a(n) = A260301(3*n). a(3*n) = A260301(n).

%e G.f. = 1 - x - 5*x^2 + 3*x^3 + 7*x^4 + 4*x^5 + 3*x^6 - 18*x^7 - 17*x^8 + ...

%t a[ n_] := SeriesCoefficient[ 2 q^(3/8) QPochhammer[ q] QPochhammer[ q^2]^4 / EllipticTheta[ 2, 0, q^(3/2)], {q, 0, n}];

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

%Y Cf. A263021, A260301.

%K sign,look

%O 0,3

%A _Michael Somos_, Jan 02 2016

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Last modified March 28 21:57 EDT 2024. Contains 371254 sequences. (Running on oeis4.)