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A123863 Expansion of (c(q^3) - c(q^6) - 2*c(q^12)) / 3 in powers of q where c(q) is a cubic AGM theta function.

%I A123863 #17 Feb 16 2025 08:33:03
%S A123863 1,-1,0,-1,0,0,2,-1,0,0,0,0,2,-2,0,-1,0,0,2,0,0,0,0,0,1,-2,0,-2,0,0,2,
%T A123863 -1,0,0,0,0,2,-2,0,0,0,0,2,0,0,0,0,0,3,-1,0,-2,0,0,0,-2,0,0,0,0,2,-2,
%U A123863 0,-1,0,0,2,0,0,0,0,0,2,-2,0,-2,0,0,2,0,0,0,0,0,0,-2,0,0,0,0,4,0,0,0,0,0,2,-3,0,-1,0,0,2,-2,0
%N A123863 Expansion of (c(q^3) - c(q^6) - 2*c(q^12)) / 3 in powers of q where c(q) is a cubic AGM theta function.
%C A123863 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%C A123863 Cubic AGM theta functions: a(q) (see A004016), b(q) (A005928), c(q) (A005882).
%H A123863 G. C. Greubel, <a href="/A123863/b123863.txt">Table of n, a(n) for n = 1..1000</a>
%H A123863 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A123863 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A123863 Expansion of (a(x) - a(x^2) - a(x^3) - 2*a(x^4) + a(x^6) + 2*a(x^12)) / 6 in powers of x where a() is a cubic AGM theta function. - _Michael Somos_, Aug 03 2015
%F A123863 Expansion of psi(-x) * psi(-x^9) * phi(x^9) / f(-x^6) in powers of x where phi(), psi(), f() are Ramanujan theta functions. - _Michael Somos_, Aug 03 2015
%F A123863 Expansion of eta(q) * eta(q^4) * eta(q^18)^4 / (eta(q^2) * eta(q^6) * eta(q^9) * eta(q^36)) in powers of q.
%F A123863 Euler transform of period 36 sequence [ -1, 0, -1, -1, -1, 1, -1, -1, 0, 0, -1, 0, -1, 0, -1, -1, -1, -2, -1, -1, -1, 0, -1, 0, -1, 0, 0, -1, -1, 1, -1, -1, -1, 0, -1, -2, ...].
%F A123863 a(n) is multiplicative with a(2^e) = -1 if e>0, a(3^e) = 0^e, a(p^e) = e+1 if p == 1 (mod 6), a(p^e) = (1 + (-1)^e)/2 if p == 5 (mod 6).
%F A123863 a(3*n) = a(6*n + 5) = 0.
%F A123863 a(2*n) = -A113448(n). a(6*n + 2) = -A033687(n).
%F A123863 a(3*n + 1) = A227696(n). a(6*n + 1) = A097195(n). a(12*n + 1) = A123884(n). a(12*n + 7) = 2 * A121361(n). - _Michael Somos_, Aug 03 2015
%e A123863 G.f. = q - q^2 - q^4 + 2*q^7 - q^8 + 2*q^13 - 2*q^14 - q^16 + 2*q^19 + ...
%t A123863 a[ n_] := If[ n < 1, 0, Times @@ (Which[ # < 4, {1, -1, 0}[[Mod[#, 3, 1]]], Mod[#, 6] == 1, #2 + 1, True, (1 + (-1)^#2) / 2] & @@@ FactorInteger @ n)]; (* _Michael Somos_, Aug 03 2015 *)
%t A123863 a[ n_] := SeriesCoefficient[ x EllipticTheta[ 2, 0, x^(9/2)] EllipticTheta[ 2, Pi/4, x^(1/2)] EllipticTheta[ 4, 0, x^18] / (2^(3/2) x^(5/4) QPochhammer[ x^6]), {x, 0, n}]; (* _Michael Somos_, Aug 03 2015 *)
%o A123863 (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^18 + A)^4 / (eta(x^2 + A) * eta(x^6 + A) * eta(x^9 + A) * eta(x^36 + A)), n))};
%o A123863 (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, p==3, 0, p%6==1, e+1, !(e%2))))};
%Y A123863 Cf. A033687, A097195, A113448, A121361, A123884.
%K A123863 sign,mult
%O A123863 1,7
%A A123863 _Michael Somos_, Oct 14 2006