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%I #40 Nov 21 2023 08:32:24
%S 1,-1,1,1,0,-1,2,-1,1,0,0,1,2,-2,0,1,0,-1,2,0,2,0,0,-1,1,-2,1,2,0,0,2,
%T -1,0,0,0,1,2,-2,2,0,0,-2,2,0,0,0,0,1,3,-1,0,2,0,-1,0,-2,2,0,0,0,2,-2,
%U 2,1,0,0,2,0,0,0,0,-1,2,-2,1,2,0,-2,2,0,1,0,0,2,0,-2,0,0,0,0,4,0,2,0,0,-1,2,-3,0,1,0,0,2,-2,0
%N Expansion of q * psi(q^3)^3 / psi(q) in powers of q where psi() is a Ramanujan theta function.
%C Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%C Cubic AGM theta functions: a(q) (see A004016), b(q) (A005928), c(q) (A005882).
%H G. C. Greubel, <a href="/A093829/b093829.txt">Table of n, a(n) for n = 1..1000</a>
%H Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>, 2019.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>.
%F Expansion of (a(q) - a(q^2)) / 6 = c(q^2)^2 / (3 * c(q)) in powers of q where a(), c() are cubic AGM functions. - _Michael Somos_, Sep 06 2007
%F Expansion of (eta(q) * eta(q^6)^6) / (eta(q^2)^2 * eta(q^3)^3) in powers of q.
%F Euler transform of period 6 sequence [ -1, 1, 2, 1, -1, -2, ...].
%F Moebius transform is period 6 sequence [ 1, -2, 0, 2, -1, 0, ...] = A112300. - _Michael Somos_, Jul 16 2006
%F Multiplicative with a(p^e) = (-1)^e if p=2; a(p^e) = 1 if p=3; a(p^e) = 1+e if p == 1 (mod 6); a(p^e) = (1 + (-1)^e) / 2 if p == 5 (mod 6).
%F G.f. is a period 1 Fourier series which satisfies f(-1 / (6 t)) = 12^(-1/2) (t/i) g(t) where q = exp(2 Pi i t) and g() is g.f. for A122859.
%F G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^4)) where f(u, v, w) = w * (u + v)^2 - v * (v + w) * (v + 4*w).
%F G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^3), A(x^6)) where f(u1, u2, u3, u6) = u2 * (u2 - u3 - 4*u6) - (u3 + u6) * (u1 - 3*u3 - 3*u6).
%F G.f.: Sum_{k>0} (x^k - 2 * x^(2*k) + 2 * x^(4*k) - x^(5*k)) / (1 - x^(6*k)) = x * Product_{k>0} ((1 - x^k) * (1 - x^(6*k))^6) / ((1 - x^(2*k))^2 * (1 - x^(3*k))^3).
%F a(n) = -(-1)^n * A113447(n). - _Michael Somos_, Jan 31 2015
%F a(2*n) = -a(n). a(3*n) = a(n). a(6*n + 5) = 0.
%F A035178(n) = |a(n)|. A033762(n) = a(2*n + 1). A033687(n) = a(3*n + 1).
%F a(4*n + 1) = A112604(n). a(4*n + 3) = A112605(n). a(6*n + 1) = A097195(n). a(8*n + 1) = A112606(n). a(8*n + 3) = A112608(n). a(8*n + 5) = 2 * A112607(n).
%F Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Pi/(6*sqrt(3)) = 0.302299894039... . - _Amiram Eldar_, Nov 21 2023
%e G.f. = q - q^2 + q^3 + q^4 - q^6 + 2*q^7 - q^8 + q^9 + q^12 + 2*q^13 + ...
%t a[ n_] := If[ n < 1, 0, DivisorSum[ n, {1, -2, 0, 2, -1, 0} [[ Mod[#, 6, 1]]] &]];
%t QP = QPochhammer; s = (QP[q]*QP[q^6]^6)/(QP[q^2]^2*QP[q^3]^3) + O[q]^105; CoefficientList[s, q] (* _Jean-François Alcover_, Nov 30 2015, adapted from PARI *)
%o (PARI) {a(n) = if( n<1, 0, polcoeff( sum( k=0, n, x^k * (1 - x^k)^2 / (1 + x^(2*k) + x^(4*k)), x * O(x^n)), n))};
%o (PARI) {a(n) = local(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( eta(x + A) * eta(x^6 + A)^6 / (eta(x^2 + A)^2 * eta(x^3 + A)^3), n))};
%o (PARI) {a(n) = if( n<1, 0, sumdiv( n, d, kronecker( -12, d) - if( d%2==0, 2 * kronecker( -3, d/2) ) ))}; /* _Michael Somos_, May 29 2005 */
%o (Sage) ModularForms( Gamma1(6), 1, prec=90).1; # _Michael Somos_, Sep 27 2013
%o (Magma) Basis( ModularForms( Gamma1(6), 1), 90) [2]; /* _Michael Somos_, Jul 02 2014 */
%Y Cf. A033687, A033762, A035178, A112300, A113447, A122859.
%Y Cf. A097195, A112604, A112605, A112606, A112607, A112608. - _Michael Somos_, Sep 27 2013
%Y Cf. A000700, A000122, A010054, A121373.
%K sign,mult,easy
%O 1,7
%A _Michael Somos_, Apr 17 2004
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