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A115660 Expansion of (phi(q) * phi(q^6) - phi(q^2) * phi(q^3)) / 2 in powers of q where phi() is a Ramanujan theta function.

%I A115660 #38 Feb 16 2025 08:33:00
%S A115660 1,-1,-1,1,-2,1,2,-1,1,2,-2,-1,0,-2,2,1,0,-1,0,-2,-2,2,0,1,3,0,-1,2,
%T A115660 -2,-2,2,-1,2,0,-4,1,0,0,0,2,0,2,0,-2,-2,0,0,-1,3,-3,0,0,-2,1,4,-2,0,
%U A115660 2,-2,2,0,-2,2,1,0,-2,0,0,0,4,0,-1,2,0,-3,0,-4,0
%N A115660 Expansion of (phi(q) * phi(q^6) - phi(q^2) * phi(q^3)) / 2 in powers of q where phi() is a Ramanujan theta function.
%C A115660 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%C A115660 Number 41 of the 74 eta-quotients listed in Table I of Martin (1996). - _Michael Somos_, Mar 14 2012
%H A115660 G. C. Greubel, <a href="/A115660/b115660.txt">Table of n, a(n) for n = 1..1000</a>
%H A115660 A. Berkovich and H. Yesilyurt, <a href="https://arxiv.org/abs/math/0611300">Ramanujan's identities and representation of integers by certain binary and quaternary quadratic forms</a>, arXiv:math/0611300 [math.NT], 2006-2007.
%H A115660 Y. Martin, <a href="http://dx.doi.org/10.1090/S0002-9947-96-01743-6">Multiplicative eta-quotients</a>, Trans. Amer. Math. Soc. 348 (1996), no. 12, 4825-4856, see page 4852 Table I.
%H A115660 Michael Somos, <a href="/A030203/a030203.txt">Index to Yves Martin's list of 74 multiplicative eta-quotients and their A-numbers</a>
%H A115660 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A115660 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A115660 Expansion of eta(q) * eta(q^4) * eta(q^6) * eta(q^24) / (eta(q^3) * eta(q^8)) in powers of q.
%F A115660 Euler transform of period 24 sequence [ -1, -1, 0, -2, -1, -1, -1, -1, 0, -1, -1, -2, -1, -1, 0, -1, -1, -1, -1, -2, 0, -1, -1, -2, ...].
%F A115660 a(n) is multiplicative with a(2^e) = a(3^e) = (-1)^e, a(p^e) = e+1 if p == 1, 7 (mod 24), a(p^e) = (e+1) * (-1)^e if p == 5, 11 (mod 24), a(p^e) = (1 + (-1)^e) / 2 if p == 13, 17, 19, 23 (mod 24).
%F A115660 G.f. is a period 1 Fourier series which satisfies f(-1 / (24 t)) = 24^(1/2) (t/i) f(t) where q = exp(2 Pi i t).
%F A115660 G.f.: Sum_{k>0} Kronecker(k,8) * x^k / (1 + x^k + x^(2*k)) = Sum_{k>0} Kronecker(k,3) * x^k * (1 - x^(2*k)) / (1 + x^(4*k)).
%F A115660 abs(a(n)) = A000377(n). a(n) = (-1)^n * A128581(n). a(2*n) = a(3*n) = -a(n). a(2*n + 1) = A128580(n). - _Michael Somos_, Mar 14 2012
%F A115660 abs(a(n)) = A192013(n) unless n=0. - _Michael Somos_, Oct 22 2015
%F A115660 a(3*n + 1) = A263571(n). a(4*n) = A259668(n). a(6*n + 1) = A261115(n). a(6*n + 4) = A263548(n). a(8*n + 1) = A260308(n). - _Michael Somos_, Oct 22 2015
%F A115660 a(n) = A000377(n) - A108563(n) = A046113(n) - A000377(n). - _Michael Somos_, Oct 22 2015
%e A115660 G.f. = q - q^2 - q^3 + q^4 - 2*q^5 + q^6 + 2*q^7 - q^8 + q^9 + 2*q^10 - 2*q^11 + ...
%t A115660 a[ n_] := SeriesCoefficient[ q QPochhammer[ q] QPochhammer[ q^4] QPochhammer[ q^6] QPochhammer[ q^24] / (QPochhammer[ q^3] QPochhammer[ q^8]), {q, 0, n}]; (* _Michael Somos_, Apr 19 2015 *)
%t A115660 a[ n_] := SeriesCoefficient[ (EllipticTheta[ 3, 0, q] EllipticTheta[ 3, 0, q^6] - EllipticTheta[ 3, 0, q^2] EllipticTheta[ 3, 0, q^3]) / 2, {q, 0, n}]; (* _Michael Somos_, Apr 19 2015 *)
%t A115660 a[ n_] := If[ n < 1, 0, Sum[ KroneckerSymbol[ 2, d] KroneckerSymbol[ -3, n/d], {d, Divisors[ n]}]]; (* _Michael Somos_, Apr 19 2015 *)
%t A115660 a[ n_] := If[ n < 1, 0, Times @@ (Which[ # == 1, 1, # < 5, (-1)^#2, Mod[#, 24] < 12, (#2 + 1) KroneckerSymbol[ #, 12]^#2, True, 1 - Mod[#2, 2]]& @@@ FactorInteger[n])]; (* _Michael Somos_, Oct 22 2015 *)
%o A115660 (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<5, (-1)^e, p%24<12, (e+1) * kronecker( p, 12)^e, 1-e%2)))};
%o A115660 (PARI) {a(n) = my(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( eta(x + A) * eta(x^4 + A) * eta(x^6 + A) * eta(x^24 + A) / (eta(x^3 + A) * eta(x^8 + A)), n))};
%o A115660 (PARI) {a(n) = if( n<1, 0, sumdiv( n, d, kronecker( 2, d) * kronecker( -3, n/d)))};
%Y A115660 Cf. A000377, A046113, A108563, A128580, A128581, A192013, A259668, A260308, A261115, A263548, A263571.
%K A115660 sign,mult
%O A115660 1,5
%A A115660 _Michael Somos_, Jan 28 2006