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A112298 Expansion of (a(q) - 3*a(q^2) + 2*a(q^4)) / 6 in powers of q where a() is a cubic AGM theta function.

%I A112298 #29 Feb 16 2025 08:32:59
%S A112298 1,-3,1,3,0,-3,2,-3,1,0,0,3,2,-6,0,3,0,-3,2,0,2,0,0,-3,1,-6,1,6,0,0,2,
%T A112298 -3,0,0,0,3,2,-6,2,0,0,-6,2,0,0,0,0,3,3,-3,0,6,0,-3,0,-6,2,0,0,0,2,-6,
%U A112298 2,3,0,0,2,0,0,0,0,-3,2,-6,1,6,0,-6,2,0,1,0,0,6,0,-6,0,0,0,0,4,0,2,0,0,-3,2,-9,0,3,0,0,2,-6,0
%N A112298 Expansion of (a(q) - 3*a(q^2) + 2*a(q^4)) / 6 in powers of q where a() is a cubic AGM theta function.
%C A112298 Cubic AGM theta functions: a(q) (see A004016), b(q) (A005928), c(q) (A005882).
%C A112298 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H A112298 G. C. Greubel, <a href="/A112298/b112298.txt">Table of n, a(n) for n = 1..1000</a>
%H A112298 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>, 2019.
%H A112298 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>.
%F A112298 From _Michael Somos_, Jan 17 2015: (Start)
%F A112298 Expansion of b(q) * (b(q^4) - b(q)) / (3*b(q^2)) in powers of q  where b() is a cubic AGM theta function.
%F A112298 Expansion of q * chi(-q)^3 * phi(-q^2) * psi(q^3) / chi(-q^6)^3 in powers of q where phi(), psi(), chi() are Ramanujan theta functions.
%F A112298 Expansion of q * phi(-q)^2 * psi(q^6)^2 / (psi(-q) * psi(-q^3)) in powers of q where phi(), psi() are Ramanujan theta functions.
%F A112298 Expansion of q * f(q) * f(-q, -q^5)^4 / f(q^3)^3 in powers of q where f() is a Ramanujan theta function. (End)
%F A112298 Expansion of (eta(q) * eta(q^12))^3 / (eta(q^2) * eta(q^3) * eta(q^4) * eta(q^6)) in powers of q.
%F A112298 Euler transform of period 12 sequence [ -3, -2, -2, -1, -3, 0, -3, -1, -2, -2, -3, -2, ...].
%F A112298 Moebius transform is period 12 sequence [ 1, -4, 0, 6, -1, 0, 1, -6, 0, 4, -1, 0, ...].
%F A112298 Multiplicative with a(2^e) = 3(-1)^e if e>0, a(3^e)=1, a(p^e) = e+1 if p == 1 (mod 6), a(p^e) = (1 + (-1)^e)/2 if p == 2 (mod 6).
%F A112298 G.f. is a period 1 Fourier series which satisfies f(-1 / (12 t)) = 12^(1/2) (t/i) f(t) where q = exp(2 Pi i t).
%F A112298 G.f.: Sum_{k>0} Kronecker(-3, k) * x^k * (1 - x^k)^2 / (1 - x^(4*k)).
%F A112298 a(n) = -(-1)^n * A244375(n). a(6*n + 5) = 0, a(3*n) = a(n).
%F A112298 a(2*n) = -3 * A093829(n). a(2*n + 1) = A033762(n). a(3*n + 1) = A129576(n). a(4*n + 1) = A112604(n). a(4*n + 3) = A112605(n). a(6*n + 1) = A097195(n). a(6*n + 2) = -3 * A033687(n).
%F A112298 Sum_{k=1..n} abs(a(k)) ~ (Pi/sqrt(3)) * n. - _Amiram Eldar_, Jan 23 2024
%e A112298 G.f. = q - 3*q^2 + q^3 + 3*q^4 - 3*q^6 + 2*q^7 - 3*q^8 + q^9 + 3*q^12 + ...
%t A112298 a[ n_] := SeriesCoefficient[ q QPochhammer[ q, q^2]^3 QPochhammer[ -q^6, q^6]^3 EllipticTheta[ 4, 0, q^2] EllipticTheta[ 2, 0, q^(3/2)] / (2 q^(3/8)), {q, 0, n}]; (* _Michael Somos_, Jan 17 2015 *)
%t A112298 a[ n_] := If[ n < 1, 0, DivisorSum[ n, JacobiSymbol[ -3, n/#] {1, -2, 1, 0}[[Mod[#, 4, 1]]] &]]; (* _Michael Somos_, Jan 17 2015 *)
%o A112298 (PARI) {a(n) = if( n<1, 0, sumdiv(n, d, kronecker(-3, n/d)*[0, 1, -2, 1][d%4 + 1]))};
%o A112298 (PARI) {a(n) = my(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( (eta(x + A) * eta(x^12 + A))^3/ (eta(x^2 + A) * eta(x^3 + A) * eta(x^4 + A) * eta(x^6 + A)), n))};
%o A112298 (Magma) A := Basis( ModularForms( Gamma1(12), 1), 106); A[2] - 3*A[3] + A[4] + 3*A[5]; /* _Michael Somos_, Jan 17 2015 */
%Y A112298 Cf. A033687, A033762, A093602, A093829, A097195, A112604, A112605, A129576, A244375.
%Y A112298 Cf. A000122, A000700, A004016, A010054, A005882, A005928, A121373.
%K A112298 sign,mult
%O A112298 1,2
%A A112298 _Michael Somos_, Sep 02 2005