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A320908 Expansion of Product_{k>=1} theta_4(x^k), where theta_4() is the Jacobi theta function.

%I A320908 #25 Sep 08 2022 08:46:23
%S A320908 1,-2,-2,2,4,6,-6,-2,-8,-12,2,6,20,14,22,-2,-14,-34,-20,-42,-48,34,10,
%T A320908 50,48,80,82,52,-16,-30,-142,-130,-138,-226,-54,-70,80,190,310,238,
%U A320908 392,178,178,86,-40,-148,-582,-506,-546,-680,-656,-126,-336,262,428,930
%N A320908 Expansion of Product_{k>=1} theta_4(x^k), where theta_4() is the Jacobi theta function.
%C A320908 Convolution of A288007 and A288098.
%C A320908 Convolution inverse of A301554.
%H A320908 Seiichi Manyama, <a href="/A320908/b320908.txt">Table of n, a(n) for n = 0..10000</a>
%F A320908 G.f.: Product_{i>=1, j>=1} (1 - x^(i*j))/(1 + x^(i*j)).
%F A320908 G.f.: Product_{k>=1} ((1 - x^k)/(1 + x^k))^d(k), where d(k) is the number of divisors of k (A000005).
%F A320908 G.f.: exp(-Sum_{k>=1} sigma(k)*x^k*(2 + x^k)/(k*(1 - x^(2*k)))).
%p A320908 with(numtheory): seq(coeff(series(mul(((1-x^k)/(1+x^k))^tau(k),k=1..n),x,n+1), x, n), n = 0 .. 60); # _Muniru A Asiru_, Oct 23 2018
%t A320908 nmax = 55; CoefficientList[Series[Product[EllipticTheta[4, 0, x^k], {k, 1, nmax}], {x, 0, nmax}], x]
%t A320908 nmax = 55; CoefficientList[Series[Product[((1 - x^k)/(1 + x^k))^DivisorSigma[0, k], {k, 1, nmax}], {x, 0, nmax}], x]
%t A320908 nmax = 55; CoefficientList[Series[Exp[-Sum[DivisorSigma[1, k] x^k (2 + x^k)/(k (1 - x^(2 k))), {k, 1, nmax}]], {x, 0, nmax}], x]
%o A320908 (PARI) N=99; x='x+O('x^N); Vec(prod(k=1, N, ((1-x^k)/(1+x^k))^numdiv(k))) \\ _Seiichi Manyama_, Oct 25 2018
%o A320908 (Magma) m:=60; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!( (&*[(&*[(1-x^(j*k))/(1+x^(j*k)):j in [1..2*m]]): k in [1..2*m]]) )); // _G. C. Greubel_, Oct 29 2018
%Y A320908 Cf. A000005, A000203, A002448, A288007, A288098, A301554, A320067, A320068.
%K A320908 sign,look
%O A320908 0,2
%A A320908 _Ilya Gutkovskiy_, Oct 23 2018