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Expansion of q^(-1/24) * eta(q)^3 / eta(q^2) in powers of q.
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%I #48 Sep 08 2022 08:45:23

%S 1,-3,1,2,2,-1,-4,1,-2,0,2,4,-1,2,-2,-1,0,-2,-2,-2,0,4,1,0,2,-2,5,0,

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

%U 1,0,0,0,4,2,0,4,0,-4,2,0,2,-1,0,0,2,-2,-2,-6,-1,2,0,0,-4,0,2,2,0,0,2,-2,2,2,0,1,0,0,2,4,0,0,-2,1,-6,0,-2,0

%N Expansion of q^(-1/24) * eta(q)^3 / eta(q^2) in powers of q.

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

%D B. Gordon and D. Sinor, Multiplicative properties of eta-products, Number theory, Madras 1987, pp. 173-200, Lecture Notes in Math., 1395, Springer, Berlin, 1989; see page 182. MR1019331 (90k:11050)

%H Seiichi Manyama, <a href="/A115110/b115110.txt">Table of n, a(n) for n = 0..1000</a>

%H George E. Andrews, <a href="http://dx.doi.org/10.1090/S0002-9947-1986-0814916-2">The fifth and seventh order mock theta functions</a>, Trans. Amer. Math. Soc., 293 (1986) 113-134; see page 124 (5.15).

%H Peter Bala, <a href="/A115110/a115110.pdf">Identities for A115110</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 f(x) * f(-x) in powers of x^2 where f() is a Ramanujan theta function.

%F Expansion of f(-x) * phi(-x) in powers of x where phi(), f() are Ramanujan theta functions.

%F Given A = A0 + A1 + A2 + A3 + A4 + A5 + A6 is the 7-section, then 0 = A0*A4 + A1*A3 + A5*A6 + 4*A2^2, A2 = x^2 * A(x^49).

%F Euler transform of period 2 sequence [ -3, -2,...].

%F G.f.: Product_{k>0} (1 - x^k)^2 / (1 + x^k).

%F G.f.: Sum_{k>=0} ( x^((3*k^2 + k)/2) * (1 - x^(2*k + 1)) * Sum_{|j|<=k} (-x)^(-j^2) ).

%F a(49*n + 2) = a(n). a(7*n + 2) = 0 unless n = 7*k.

%F a(n) = (-1)^n * A107033(n).

%F G.f.: exp( Sum_{n>=1} -sigma(2*n)*x^n/n ). - _Seiichi Manyama_, Mar 02 2017

%F a(n) = -(1/n)*Sum_{k=1..n} sigma(2*k)*a(n-k). - _Seiichi Manyama_, Mar 04 2017

%F From _Peter Bala_, Jan 01 2021: (Start)

%F For prime p of the form 4*k + 3, a(n*p^2 + (p^2 - 1)/24) = e*a(n), where e = 1 if p == 7 or 23 (mod 24) and e = -1 if p == 11 or 19 (mod 24).

%F If n > 0 and p are coprime then a(n*p + (p^2 - 1)/24) = 0. Cf. A002107.

%F (End)

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

%e G.f. = q - 3*q^25 + q^49 + 2*q^73 + 2*q^97 - q^121 - 4*q^145 + q^169 - 2*q^193 + ...

%p prod := n -> mul( (1 - x^k)^2*(1 - x^(2*k-1)), k = 1..n):

%p a := n -> coeff(prod(100), x, n):

%p seq(a(n), n = 0..100); # _Peter Bala_, Jan 01 2021

%t a[ n_] := SeriesCoefficient[ QPochhammer[ x]^3 / QPochhammer[ x^2], {x, 0, n}]; (* _Michael Somos_, Jul 12 2012 *)

%t a[ n_] := SeriesCoefficient[ QPochhammer[ x] QPochhammer[ -x], {x, 0, 2 n}]; (* _Michael Somos_, Jul 12 2012 *)

%t a[ n_] := SeriesCoefficient[ QPochhammer[ x] EllipticTheta[ 4, 0, x], {x, 0, n}]; (* _Michael Somos_, Jul 12 2012 *)

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

%o (Magma) m:=120; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!( (&*[(1 - x^j)^2 / (1 + x^j): j in [1..m+2]]) )); // _G. C. Greubel_, Nov 18 2018

%o (Sage)

%o R = PowerSeriesRing(ZZ, 'x')

%o x = R.gen().O(120)

%o s = prod((1 - x^j)^2 / (1 + x^j) for j in (1..120))

%o s.coefficients() # _G. C. Greubel_, Nov 18 2018

%Y Cf. A107033, A002107.

%Y Cf. Product_{n>=1} (1 - q^n)^(k+1)/(1 - q^(k*n)): A010815 (k=1), this sequence (k=2), A185654 (k=3), A282937 (k=5), A282942 (k=7).

%K sign,easy

%O 0,2

%A _Michael Somos_, Mar 07 2006