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A132069 Expansion of eta(q) * eta(q^2)^2 * eta(q^5)^3 / eta(q^10)^2 in powers of q.

%I A132069 #26 Feb 16 2025 08:33:06
%S A132069 1,-1,-3,2,1,-1,6,6,-7,-7,-3,-12,-2,12,18,2,9,16,-21,-20,1,-12,-36,22,
%T A132069 14,-1,36,20,-6,-30,6,-32,-23,24,48,6,7,36,-60,-24,-7,-42,-36,42,12,
%U A132069 -7,66,46,-18,-43,-3,-32,-12,52,60,-12,42,40,-90,-60,-2,-62,-96,42,41,12,72,66,-16,-44,18,-72,-49,72,108,2,20,72
%N A132069 Expansion of eta(q) * eta(q^2)^2 * eta(q^5)^3 / eta(q^10)^2 in powers of q.
%C A132069 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%C A132069 Denoted by z(q) = q d/dq log k(q) in Cooper (2009) where k() is the g.f. of A112274. - _Michael Somos_, Jul 08 2012
%D A132069 Bruce C. Berndt, Ramanujan's Notebooks Part III, Springer-Verlag, 1991, see p. 253 Eq. (8.12)
%H A132069 G. C. Greubel, <a href="/A132069/b132069.txt">Table of n, a(n) for n = 0..10000</a>
%H A132069 Shaun Cooper, <a href="https://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. 312, eq. (1.4).
%H A132069 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>, 2019.
%H A132069 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>.
%F A132069 Expansion of (5 * phi(-q) * phi(-q^5)^3 - phi(-q)^3 * phi(-q^5)) / 4 in powers of q where phi() is a Ramanujan theta function.
%F A132069 Euler transform of period 10 sequence [-1, -3, -1, -3, -4, -3, -1, -3, -1, -4, ...].
%F A132069 a(n) = -b(n) where b() is multiplicative with b(5^e) = 1, b(2^e) = 2 - ((-2)^(e+1) - 1) / (-2 - 1), b(p^e) = (p^(e+1) - 1) / (p - 1) if p == 1, 9 (mod 10), b(p^e) = ((-p)^(e+1) - 1) / (-p - 1) if p == 3, 7 (mod 10).
%F A132069 G.f.: Product_{k>0} (1 - x^k) * (1 - x^(2*k))^2 * (1 - x^(5*k)) / (1 + x^(5*k))^2.
%F A132069 G.f.: 1 + Sum_{k>0} (-1)^k * k * x^k / (1 - x^k) * Kronecker(5, k).
%F A132069 G.f. is a period 1 Fourier series which satisfies f(-1 / (10 t)) = 2000^(1/2) (t/i)^2 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A129303.
%F A132069 a(n) = (-1)^n * A113185(n).
%F A132069 Sum_{k=1..n} abs(a(k)) ~ c * n^2, where c = Pi^2/(12*sqrt(5)) = 0.367818... . - _Amiram Eldar_, Jan 28 2024
%e A132069 G.f. = 1 - q - 3*q^2 + 2*q^3 + q^4 - q^5 + 6*q^6 + 6*q^7 - 7*q^8 - 7*q^9 +...
%t A132069 a[ n_] := If[ n < 1, Boole[n == 0], DivisorSum[ n, KroneckerSymbol[ 5, #] # (-1)^# &]]; (* _Michael Somos_, Aug 26 2015 *)
%t A132069 a[ n_] := SeriesCoefficient[ QPochhammer[ q] QPochhammer[ q^2]^2 QPochhammer[ q^5]^3 / QPochhammer[ q^10]^2, {q, 0, n}]; (* _Michael Somos_, Aug 26 2015 *)
%t A132069 a[ n_] := SeriesCoefficient[ (5 EllipticTheta[ 4, 0, q] EllipticTheta[ 4, 0, q^5]^3 - EllipticTheta[ 4, 0, q]^3 EllipticTheta[ 4, 0, q^5])/4, {q, 0, n}]; (* _Michael Somos_, Aug 26 2015 *)
%o A132069 (PARI) {a(n) = if( n<1, n==0, sumdiv( n, d, kronecker(5, d) * d * (-1)^d))};
%o A132069 (PARI) {a(n) = my(A, p, e, a1); if( n<1, n==0, A = factor(n); -prod( k=1, matsize(A)[1], [p, e] = A[k, ]; if( p==5, 1, p>2, p *= kronecker(5, p); (p^(e+1) - 1) / (p - 1), (5 + (-2)^(e+1)) / 3)))};
%o A132069 (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A) * eta(x^2 + A)^2 * eta(x^5 + A)^3 / eta(x^10 + A)^2, n))};
%Y A132069 Cf. A112274, A113185, A129303.
%Y A132069 Cf. A000122, A000700, A010054, A121373.
%K A132069 sign
%O A132069 0,3
%A A132069 _Michael Somos_, Aug 08 2007, Mar 20 2008