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A244600 Expansion of f(-x) / f(-x^7) in powers of x where f() is a Ramanujan theta function.

%I A244600 #17 May 06 2025 11:15:37
%S A244600 1,-1,-1,0,0,1,0,2,-1,-1,0,0,0,0,3,-3,-2,0,0,1,0,5,-3,-3,0,0,2,0,8,-6,
%T A244600 -5,0,0,3,0,11,-8,-7,0,0,3,0,17,-13,-11,0,0,6,0,24,-17,-14,0,0,7,0,34,
%U A244600 -25,-21,0,0,11,0,47,-33,-28,0,0,14,0,64,-47,-39
%N A244600 Expansion of f(-x) / f(-x^7) in powers of x where f() is a Ramanujan theta function.
%H A244600 Seiichi Manyama, <a href="/A244600/b244600.txt">Table of n, a(n) for n = 0..10000</a>
%H A244600 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A244600 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A244600 Expansion of q^(1/4) * eta(q) / eta(q^7) in powers of q.
%F A244600 Euler transform of period 7 sequence [ -1, -1, -1, -1, -1, -1, 0, ...].
%F A244600 Given g.f. A(x), then B(q) = A(q^4) / q satisfies 0 = f(B(q), B(q^3)) where f(u, v) = (u - v^3) * (u^3 - v) - 3*u*v * (2 + u*v).
%F A244600 G.f. is a period 1 Fourier series which satisfies f(-1 / (112 t)) = 7^(1/2) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A035985.
%F A244600 G.f.: Product_{k>0} (1 - x^k) / (1 - x^(7*k)).
%F A244600 Convolution inverse of A035985.
%F A244600 a(7*n + 3) = a(7*n + 4) = a(7*n + 6) = 0.
%F A244600 a(n) = -(1/n)*Sum_{k=1..n} A113957(k)*a(n-k), a(0) = 1. - _Seiichi Manyama_, Mar 25 2017
%e A244600 G.f. = 1 - x - x^2 + x^5 + 2*x^7 - x^8 - x^9 + 3*x^14 - 3*x^15 - 2*x^16 + ...
%e A244600 G.f. = q^-1 - q^3 - q^7 + q^19 + 2*q^27 - q^31 - q^35 + 3*q^55 - 3*q^59 + ...
%t A244600 a[ n_] := SeriesCoefficient[ QPochhammer[ x] / QPochhammer[ x^7], {x, 0, n}];
%o A244600 (PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A) / eta(x^7 + A), n))};
%Y A244600 Cf. A035985.
%K A244600 sign
%O A244600 0,8
%A A244600 _Michael Somos_, Jul 01 2014