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A110712 Expansion of eta(q)^3 * eta(q^5) * eta(q^10)^2 / eta(q^2)^2 in powers of q.

%I A110712 #26 Feb 16 2025 08:32:58
%S A110712 1,-3,2,-1,5,-6,6,-7,7,-15,12,-2,12,-18,10,-9,16,-21,20,-5,12,-36,22,
%T A110712 -14,25,-36,20,-6,30,-30,32,-23,24,-48,30,-7,36,-60,24,-35,42,-36,42,
%U A110712 -12,35,-66,46,-18,43,-75,32,-12,52,-60,60,-42,40,-90,60,-10,62,-96,42,-41,60,-72,66,-16,44,-90,72,-49,72,-108
%N A110712 Expansion of eta(q)^3 * eta(q^5) * eta(q^10)^2 / eta(q^2)^2 in powers of q.
%C A110712 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H A110712 G. C. Greubel, <a href="/A110712/b110712.txt">Table of n, a(n) for n = 1..1000</a>
%H A110712 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. 318, Th. 4.1.
%H A110712 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>, 2019.
%H A110712 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>.
%F A110712 Expansion of (phi(-q) * phi(-q^5)^3 - phi(-q)^3 * phi(-q^5)) / 4 in powers of q where phi() is a Ramanujan theta function. - _Michael Somos_, Jul 12 2012
%F A110712 Euler transform of period 10 sequence [-3, -1, -3, -1, -4, -1, -3, -1, -3, -4, ...].
%F A110712 Multiplicative with a(p^e) = p^e if p=5, a(p^e) = -(p^(e+1) - 5*(-1)^e) / (p + 1) if p=2, a(p^e) = (p^(e+1) - 1) / (p - 1) if p == 1 or 9 (mod 10), a(p^e) = (p^(e+1) - (-1)^e) / (p + 1) if p == 3 or 7 (mod 10).
%F A110712 G.f.: Sum_{k>0} Kronecker(k, 5) * x^k / (1 + x^k)^2 = x * Product_{k>0} (1 - x^k)^3 * (1 - x^(5*k)) * (1 - x^(10*k))^2 / (1 - x^(2*k))^2.
%F A110712 a(n) = (-1)^(n+1) * A138483(n). - _Amiram Eldar_, Jan 28 2024
%e A110712 G.f. = q - 3*q^2 + 2*q^3 - q^4 + 5*q^5 - 6*q^6 + 6*q^7 - 7*q^8 + 7*q^9 - 15*q^10 + ...
%t A110712 a[ n_] := If[ n < 1, 0, DivisorSum[ n, -(-1)^# # KroneckerSymbol[ 5, n/#] &]]; (* _Michael Somos_, Jul 12 2012 *)
%t A110712 a[ n_] := SeriesCoefficient[ (1/4) (EllipticTheta[ 4, 0, q] EllipticTheta[ 4, 0, q^5]^3 - EllipticTheta[ 4, 0, q]^3 EllipticTheta[ 4, 0, q^5]), {q, 0, n}]; (* _Michael Somos_, Jul 12 2012 *)
%o A110712 (PARI) {a(n) = if( n<1, 0, sumdiv( n, d, -(-1)^d * d * kronecker( n/d, 5)))};
%o A110712 (PARI) {a(n) = my(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( eta(x + A)^3 * eta(x^5 + A) * eta(x^10 + A)^2 / eta(x^2 + A)^2, n))};
%Y A110712 Cf. A138483.
%Y A110712 Cf. A000122, A000700, A010054, A121373.
%K A110712 sign,mult
%O A110712 1,2
%A A110712 _Michael Somos_, Aug 05 2005