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A140727 Expansion of (phi(q) * phi(q^15) - phi(q^3) * phi(q^5)) / 2 in powers of q where phi() is a Ramanujan theta function.

%I A140727 #15 Feb 16 2025 08:33:08
%S A140727 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 A140727 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 A140727 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 A140727 Expansion of (phi(q) * phi(q^15) - phi(q^3) * phi(q^5)) / 2 in powers of q where phi() is a Ramanujan theta function.
%C A140727 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%D A140727 B. C. Berndt, Ramanujan's Notebooks Part III, Springer-Verlag, see p. 377, Entry 9(v).
%H A140727 G. C. Greubel, <a href="/A140727/b140727.txt">Table of n, a(n) for n = 1..5000</a>
%H A140727 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A140727 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A140727 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 A140727 Expansion of eta(q^2) * eta(q^6)^2 * eta(q^10)^2 * eta(q^30) / (eta(q^3) * eta(q^5) * eta(q^12) * eta(q^20)) in powers of q.
%F A140727 Euler transform of period 60 sequence [ 0, -1, 1, -1, 1, -2, 0, -1, 1, -2, 0, -1, 0, -1, 2, -1, 0, -2, 0, -1, 1, -1, 0, -1, 1, -1, 1, -1, 0, -4, 0, -1, 1, -1, 1, -1, 0, -1, 1, -1, 0, -2, 0, -1, 2, -1, 0, -1, 0, -2, 1, -1, 0, -2, 1, -1, 1, -1, 0, -2, ...].
%F A140727 a(n) is multiplicative with a(2^e) = (-1)^e * (e-1) 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 A140727 G.f. is a period 1 Fourier series which satisfies f(-1 / (60 t)) = 60^(1/2) (t/i) f(t) where q = exp(2 Pi i t).
%F A140727 a(n) = -(-1)^n * A140728(n). abs(a(n)) = A122855(n).
%e A140727 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 A140727 a[ n_] := If[ n < 1, 0, DivisorSum[ n, (-1)^(n + #) KroneckerSymbol[ 5, #] KroneckerSymbol[ -3, n/#] &]]; (* _Michael Somos_, Aug 26 2015 *)
%t A140727 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 *)
%o A140727 (PARI) {a(n) = if( n<1, 0, sumdiv(n, d, (-1)^(n + d) * kronecker(5, d) * kronecker(-3, n/d)))}
%o A140727 (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 * (e-1), p==3 || p==5, (-1)^e, kronecker(p,15)==1, (e+1) * (-1)^(e*valuation(p%15, 2)), (1 + (-1)^e) / 2)))};
%o A140727 (PARI) {a(n) = my(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( eta(x^2 + A) * eta(x^6 + A)^2 * eta(x^10 + A)^2 * eta(x^30 + A) / (eta(x^3 + A) * eta(x^5 + A) * eta(x^12 + A) * eta(x^20 + A)), n))};
%Y A140727 Cf. A122855, A140728.
%K A140727 sign,mult
%O A140727 1,8
%A A140727 _Michael Somos_, May 29 2008