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A279918 Expansion of f(-x^2)^7 / (f(x) * f(-x^8)^2) in powers of x where f() is a Ramanujan theta function. 1

%I

%S 1,-1,-5,4,5,0,11,-15,-18,3,-10,29,10,11,37,-51,-16,-30,-65,62,53,22,

%T 50,-61,-52,-4,-81,120,62,0,124,-182,-85,-43,-157,171,123,60,202,-198,

%U -174,0,-190,301,117,54,278,-375,-171,-153,-399,370,300,108,408,-451

%N Expansion of f(-x^2)^7 / (f(x) * f(-x^8)^2) in powers of x where f() 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="/A279918/b279918.txt">Table of n, a(n) for n = 0..1000</a>

%H M. Somos, <a href="http://somos.crg4.com/multiq.html">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)^3 * chi(x^2)^2 * f(-x)^4 in powers of x where chi(), f() are Ramanujan theta functions.

%F Expansion of phi(-x^2)^4 * chi(x^2)^2 / chi(x) in powers of x where chi(), phi() are Ramanujan theta functions.

%F Expansion of q^(-1/8) * eta(q) * eta(q^2)^4 * eta(q^4) / eta(q^8)^2 in powers of q.

%F Euler transform of period 8 sequence [ -1, -5, -1, -6, -1, -5, -1, -4, ...].

%F a(n) = A279955(2*n).

%e G.f. = 1 - x - 5*x^2 + 4*x^3 + 5*x^4 + 11*x^6 - 15*x^7 - 18*x^8 + ...

%e G.f. = q^-1 - q^7 - 5*q^15 + 4*q^23 + 5*q^31 + 11*q^47 - 15*q^55 + ...

%t a[ n_] := SeriesCoefficient[ QPochhammer[ -x, x^2]^3 QPochhammer[ -x^2, x^4]^2 QPochhammer[ x]^4, {x, 0, n}];

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

%o (PARI) lista(nn) = {q='q+O('q^nn); Vec(eta(q)*eta(q^2)^4*eta(q^4)/eta(q^8)^2)} \\ _Altug Alkan_, Mar 21 2018

%Y Cf. A279955.

%K sign

%O 0,3

%A _Michael Somos_, Dec 23 2016

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Last modified October 23 14:29 EDT 2019. Contains 328345 sequences. (Running on oeis4.)