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A111580 Expansion of eta(q)^2 * eta(q^2) * eta(q^10)^3 / eta(q^5)^2 in powers of q.

%I A111580 #35 Feb 16 2025 08:32:58
%S A111580 1,-2,-2,4,1,4,-6,-8,7,-2,12,-8,-12,12,-2,16,-16,-14,20,4,12,-24,-22,
%T A111580 16,1,24,-20,-24,30,4,32,-32,-24,32,-6,28,-36,-40,24,-8,42,-24,-42,48,
%U A111580 7,44,-46,-32,43,-2,32,-48,-52,40,12,48,-40,-60,60,-8,62,-64,-42,64,-12,48,-66,-64,44,12,72,-56,-72,72
%N A111580 Expansion of eta(q)^2 * eta(q^2) * eta(q^10)^3 / eta(q^5)^2 in powers of q.
%C A111580 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%D A111580 Bruce C. Berndt, Ramanujan's Notebooks Part III, Springer-Verlag, 1991, see p. 249, Entry 8(i).
%H A111580 G. C. Greubel, <a href="/A111580/b111580.txt">Table of n, a(n) for n = 1..10000</a>
%H A111580 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. 318, Th. 4.1.
%H A111580 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>, 2019.
%H A111580 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>.
%F A111580 Expansion of q * psi(q)^3 * psi(q^5) - 5*q^2 * psi(q) * psi(q^5)^3 in powers of q where psi() is a Ramanujan theta function.
%F A111580 Euler transform of period 10 sequence [-2, -3, -2, -3, 0, -3, -2, -3, -2, -4, ...].
%F A111580 G.f.: Sum_{k>0} Kronecker(k, 5) * k * x^k / (1 - x^(2*k)) = x * Product_{k>0} (1 - x^k)^2 * (1 - x^(2*k)) * (1 + x^(5*k))^2 * (1 - x^(10*k)).
%F A111580 a(2*n) = -2*a(n).
%F A111580 From _Amiram Eldar_, Jan 28 2024: (Start)
%F A111580 a(n) = (-1)^(n+1) * A138558(n).
%F A111580 Sum_{k=1..n} abs(a(k)) ~ c * n^2, where c = Pi^2/(12*sqrt(5)) =  0.367818... . (End)
%e A111580 G.f. = q - 2*q^2 - 2*q^3 + 4*q^4 + q^5 + 4*q^6 - 6*q^7 - 8*q^8 + 7*q^9 - 2*q^10 + ...
%t A111580 a[n_] := Sum[Mod[n/d, 2]*d*KroneckerSymbol[d, 5], {d, Divisors[n]}]; Table[a[n], {n, 1, 74}](* _Jean-François Alcover_, May 11 2012, after PARI *)
%t A111580 a[ n_] := SeriesCoefficient[ (1/16) (EllipticTheta[ 2, 0, q]^3 EllipticTheta[ 2, 0, q^5] - 5 EllipticTheta[ 2, 0, q] EllipticTheta[ 2, 0, q^5]^3), {q, 0, 2 n}]; (* _Michael Somos_, Jul 12 2012 *)
%o A111580 (PARI) {a(n) = if( n<1, 0, sumdiv( n, d, (n/d%2) * d * kronecker( d, 5)))};
%o A111580 (PARI) {a(n) = my(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( eta(x + A)^2 * eta(x^2 + A) * eta(x^10 + A)^3 / eta(x^5 + A)^2, n))};
%o A111580 (PARI) q='q+O('q^99); Vec(eta(q)^2*eta(q^2)*eta(q^10)^3/eta(q^5)^2) \\ _Altug Alkan_, Apr 18 2018
%Y A111580 Cf. A138558.
%Y A111580 Cf. A000122, A000700, A010054, A121373.
%K A111580 sign,mult
%O A111580 1,2
%A A111580 _Michael Somos_, Aug 08 2005