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%I #26 Mar 12 2021 22:24:42
%S 1,-1,0,-1,1,0,0,-1,1,-1,0,0,2,0,0,-1,2,-1,0,-1,0,0,0,0,1,-2,0,0,2,0,
%T 0,-1,0,-2,0,-1,2,0,0,-1,2,0,0,0,1,0,0,0,1,-1,0,-2,2,0,0,0,0,-2,0,0,2,
%U 0,0,-1,2,0,0,-2,0,0,0,-1,2,-2,0,0,0,0,0,-1,1,-2,0,0,2,0,0,0,2,-1,0,0,0,0,0,0,2,-1,0,-1,2,0,0,-2
%N Expansion of (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).
%H G. C. Greubel, <a href="/A094247/b094247.txt">Table of n, a(n) for n = 1..1000</a>
%H S. 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. (2.4).
%H L.-C. 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. 336 eq. (3.13), p. 338 eq. (3.21).
%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 q * f(q^5) * f(-q^20) * chi(-q) in powers of q where f() and chi() are Ramanujan theta functions.
%F Expansion of eta(q) * eta(q^10)^3 / (eta(q^2) * eta(q^5)) in powers of q.
%F Euler transform of period 10 sequence [-1, 0, -1, 0, 0, 0, -1, 0, -1, -2, ...].
%F a(n) is multiplicative with a(2^e) = -1 if e > 0. a(5^e) = 1, a(p^e) = e+1 if p == 1, 5 (mod 8), a(p^e) = (1 + (-1)^e) / 2 if p == 3, 7 (mod 8).
%F G.f. is a period 1 Fourier series which satisfies f(-1 / (40 t)) = 4 (t/i) g(t) where q = exp(2 Pi i t) and g(t) is the g.f. for A214316. - _Michael Somos_, Jul 12 2012
%F G.f.: x * Product_{k>0} (1 - x^k) * (1 - x^(10*k))^3 / ((1 - x^(2*k)) * (1 - x^(5*k))).
%F G.f.: Sum_{k>0} Kronecker( -100, k) * x^k / (1 + x^k) = Sum_{k>0} Kronecker( -25, k) * x^k * (1 - x^k)^2 / (1 - x^(4*k)). - _Michael Somos_, Jul 12 2012
%F a(n+1) = (-1)^n * A053694(n). a(4*n + 1) = A122190(n).
%F a(4*n + 3) = 0. a(2*n) = - A053694(n). - _Michael Somos_, Jul 12 2012
%e G.f. = q - q^2 - q^4 + q^5 - q^8 + q^9 - q^10 + 2*q^13 - q^16 + 2*q^17 - q^18 - q^20 + ...
%t a[ n_] := SeriesCoefficient[ (EllipticTheta[ 4, 0, q^5]^2 - EllipticTheta[ 4, 0, q]^2)/4, {q, 0, n}]; (* _Michael Somos_, Jul 12 2012 *)
%t a[ n_] := SeriesCoefficient[ q QPochhammer[ -q^5] QPochhammer[ q^20] QPochhammer[q, q^2], {q, 0, n}]; (* _Michael Somos_, Jul 12 2012 *)
%o (PARI) {a(n) = my(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( eta(x + A) * eta(x^10 + A)^3 / (eta(x^2 + A) * eta(x^5 + A)),n))};
%o (PARI) {a(n) = if( n<1, 0, -(-1)^n * sumdiv( n, d, kronecker( -100, d)))}; /* _Michael Somos_, Aug 24 2006 */
%Y Cf. A053694, A122190, A214316.
%K sign,mult
%O 1,13
%A _Michael Somos_, Apr 24 2004
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