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A140728 Expansion of (phi(-q^3) * phi(-q^5) - phi(-q) * phi(-q^15)) / 2 in powers of q where phi() is a Ramanujan theta function.

%I A140728 #19 Feb 16 2025 08:33:08
%S A140728 1,0,-1,-1,-1,0,0,2,1,0,0,1,0,0,1,-3,-2,0,2,1,0,0,-2,-2,1,0,-1,0,0,0,
%T A140728 2,4,0,0,0,-1,0,0,0,-2,0,0,0,0,-1,0,-2,3,1,0,2,0,-2,0,0,0,-2,0,0,-1,2,
%U A140728 0,0,-5,0,0,0,2,2,0,0,2,0,0,-1,-2,0,0,2,3,1,0,-2,0,2,0,0,0,0,0,0,2,-2,0,-2,-4,0,0,0,-1,0,0,0,0,0
%N A140728 Expansion of (phi(-q^3) * phi(-q^5) - phi(-q) * phi(-q^15)) / 2 in powers of q where phi() is a Ramanujan theta function.
%C A140728 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H A140728 G. C. Greubel, <a href="/A140728/b140728.txt">Table of n, a(n) for n = 1..10000</a>
%H A140728 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A140728 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A140728 Expansion of q * f(-q^2) * f(-q^30) * chi(-q^3) * chi(-q^5) in powers of q where f(), chi() are Ramanujan theta functions.
%F A140728 Expansion of eta(q^2) * eta(q^3) * eta(q^5) * eta(q^30) / (eta(q^6) * eta(q^10)) in powers of q.
%F A140728 Euler transform of period 30 sequence [0, -1, -1, -1, -1, -1, 0, -1, -1, -1, 0, -1, 0, -1, -2, -1, 0, -1, 0, -1, -1, -1, 0, -1, -1, -1, -1, -1, 0, -2, ...].
%F A140728 a(n) is multiplicative with a(2^e) = (-1)^e * (1-e) if e > 0. a(3^e) = a(5^e) = (-1)^e, a(p^e) = e+1 if p == 1, 4 (mod 15), a(p^e) = (-1)^e * (e+1) if p == 2, 8 (mod 15), a(p^e) = (1 + (-1)^e) / 2 if p == 7, 11, 13, 14 (mod 15).
%F A140728 G.f. is a period 1 Fourier series which satisfies f(-1 / (30 t)) = 60^(1/2) (t/i) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A121362.
%F A140728 G.f.: x * Product_{k>0} (1 - x^(2*k)) * (1 - x^(30*k)) / ((1 + x^(3*k)) * (1 + x^(5*k))).
%F A140728 G.f.: Sum_{k>0} Kronecker(5, n) * x^n / (1 - x^n + x^(2*n)) = Sum_{k>0} -(-1)^n * Kronecker(5, n) * x^n / (1 + x^n + x^(2*n)).
%F A140728 a(n) = -(-1)^n * A140727(n). abs(a(n)) = A122855(n).
%e A140728 G.f. = q - q^3 - q^4 - q^5 + 2*q^8 + q^9 + q^12 + q^15 - 3*q^16 - 2*q^17 + ...
%t A140728 a[ n_] := If[ n < 1, 0, DivisorSum[ n, -(-1)^# KroneckerSymbol[ 5, #] KroneckerSymbol[ -3, n/#] &]]; (* _Michael Somos_, Aug 26 2015 *)
%t A140728 a[ n_] := SeriesCoefficient[ q QPochhammer[ q^2] QPochhammer[ q^30] QPochhammer[ q^3, q^6] QPochhammer[ q^5, q^10], {q, 0, n}]; (* _Michael Somos_, Aug 26 2015 *)
%t A140728 a[ n_] := SeriesCoefficient[ (EllipticTheta[ 4, 0, q^3] EllipticTheta[ 4, 0, q^5] - EllipticTheta[ 4, 0, q] EllipticTheta[ 4, 0, q^15]) / 2, {q, 0, n}]; (* _Michael Somos_, Aug 26 2015 *)
%o A140728 (PARI) {a(n) = if( n<1, 0, sumdiv(n, d, -(-1)^d * kronecker(5, d) * kronecker(-3, n/d)))};
%o A140728 (PARI) {a(n) = my(A, p, e); if( n<1, 0, A = factor(n); prod( k=1, matsize(A)[1], [p, e] = A[k, ]; if( p==2, (-1)^e * (1-e), p==3 || p==5, (-1)^e, kronecker(p, 15)==1, (e+1) * (-1)^(e*valuation(p%15, 2)), (1 + (-1)^e) / 2)))};
%o A140728 (PARI) {a(n) = my(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( eta(x^2 + A) * eta(x^3 + A) * eta(x^5 + A) * eta(x^30 + A) / (eta(x^6 + A) * eta(x^10 + A)), n))};
%Y A140728 Cf. A122855, A140727.
%K A140728 sign,mult
%O A140728 1,8
%A A140728 _Michael Somos_, May 29 2008