%I #27 Jan 27 2024 07:07:20

%S 1,1,-1,0,-1,-3,0,0,-1,1,3,0,0,2,0,0,-1,2,-1,0,3,0,0,0,0,-7,-2,0,0,2,

%T 0,0,-1,0,-2,0,-1,2,0,0,3,2,0,0,0,-3,0,0,0,1,7,0,-2,2,0,0,0,0,-2,0,0,

%U 2,0,0,-1,-6,0,0,-2,0,0,0,-1,2,-2,0,0,0,0,0,3,1,-2,0,0,-6,0,0,0,2,3,0,0,0,0,0,0,2,-1,0,7,2,0,0,-2

%N Expansion of ( 5 * phi(-q^5)^2 - phi(-q)^2 ) / 4 in powers of q where phi() is a Ramanujan theta function.

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

%C Multiplicative because this sequence is the inverse Moebius transform of a multiplicative sequence. - _Andrew Howroyd_, Aug 06 2018

%H G. C. Greubel, <a href="/A133573/b133573.txt">Table of n, a(n) for n = 0..1000</a>

%H Shaun Cooper, <a href="https://dx.doi.org/10.1007/s11139-009-9198-5">On Ramanujan's function k(q)=r(q)r^2(q^2)</a>, Ramanujan J., 20 (2009), 311-328; see p. 313, eq. (2.5).

%H Li-Chien Shen, <a href="https://doi.org/10.1090/S0002-9947-1994-1250827-3">On the additive formulas of the theta functions and a collection of Lambert series pertaining to the modular equations of degree 5</a>, Trans. Amer. Math. Soc. 345 (1994), no. 1, 323-345; see p. 335, eq. (3.5), p. 342, eq. (3.42).

%H Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>, 2019.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>.

%F Expansion of eta(q^2)^3 * eta(q^5) / ( eta(q) * eta(q^10) ) in powers of q.

%F Euler transform of period 10 sequence [ 1, -2, 1, -2, 0, -2, 1, -2, 1, -2, ...].

%F Moebius transform is period 40 sequence [ 1, -2, -1, 0, -4, 2, -1, 0, 1, 8, -1, 0, 1, 2, 4, 0, 1, -2, -1, 0, 1, 2, -1, 0, -4, -2, -1, 0, 1, -8, -1, 0, 1, -2, 4, 0, 1, 2, -1, 0, ...].

%F G.f. is a period 1 Fourier series which satisfies f(-1 / (40 t)) = 20 (t/i) g(t) where q = exp(2 Pi i t) and g() is g.f. for A122190.

%F a(n) = (-1)^n * A133574(n). a(2*n) = A133574(n). a(4*n + 1) = A214316(n). a(4*n + 3) = a(9*n + 3) = a(9*n + 6) = 0. a(9*n) = a(n). - _Michael Somos_, Jul 12 2012

%F Sum_{k=1..n} abs(a(k)) ~ (8*Pi/25) * n. - _Amiram Eldar_, Jan 27 2024

%e 1 + q - q^2 - q^4 - 3*q^5 - q^8 + q^9 + 3*q^10 + 2*q^13 - q^16 + ...

%t a[ n_] := SeriesCoefficient[ (5 EllipticTheta[ 4, 0, q^5]^2 - EllipticTheta[ 4, 0, q]^2)/4, {q, 0, n}] (* _Michael Somos_, Jul 12 2012 *)

%t a[ n_] := SeriesCoefficient[ QPochhammer[ q^2]^2 QPochhammer[ q^5, q^10] / QPochhammer[ q, q^2], {q, 0, n}] (* _Michael Somos_, Jul 12 2012 *)

%o (PARI) {a(n) = if( n<1, n==0, (-1)^n * sumdiv(n, d, if( d%5==0, kronecker(-4, d/5) * 5) - kronecker(-4, d)))}

%o (PARI) {a(n) = local(A); if( n<0, 0, A = x*O(x^n); polcoeff( eta(x^2 + A)^3 * eta(x^5+A) / (eta(x + A) * eta(x^10 + A)), n))}

%Y Cf. A122190, A133574, A214316.

%Y Cf. A000700, A000122, A010054, A121373.

%K sign,mult

%O 0,6

%A _Michael Somos_, Sep 17 2007