A253623 Expansion of phi(q) * f(q, q^2)^2 / f(q^2, q^4) in powers of q where phi(), f() are Ramanujan theta functions.

1, 4, 6, 4, 0, 0, 6, 8, 6, 4, 0, 0, 0, 8, 12, 0, 0, 0, 6, 8, 0, 8, 0, 0, 6, 4, 12, 4, 0, 0, 0, 8, 6, 0, 0, 0, 0, 8, 12, 8, 0, 0, 12, 8, 0, 0, 0, 0, 0, 12, 6, 0, 0, 0, 6, 0, 12, 8, 0, 0, 0, 8, 12, 8, 0, 0, 0, 8, 0, 0, 0, 0, 6, 8, 12, 4, 0, 0, 12, 8, 0, 4, 0, 0

Offset: 0

Author

Michael Somos, Jan 06 2015

Keywords

• nonn
• easy

Comments

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

Cubic AGM theta functions: a(q) (see A004016), b(q) (A005928), c(q) (A005882).

Examples

			G.f. = 1 + 4*x + 6*x^2 + 4*x^3 + 6*x^6 + 8*x^7 + 6*x^8 + 4*x^9 + 8*x^13 + ...
		

Links

• Michael Somos, Introduction to Ramanujan theta functions, 2019.
• Eric Weisstein's World of Mathematics, Ramanujan Theta Functions.

Crossrefs

Cf. A004016, A033687, A033762, A097195, A112604, A112605, A121361, A123884, A186706, A244339, A253625.

Cf. A000122, A000700, A005882, A005928, A010054, A121373.

Programs

Magma

A := Basis( ModularForms( Gamma1(12), 1), 83); A[1] + 4*A[2] + 6*A[3] + 4*A[4];

Mathematica

a[ n_] := SeriesCoefficient[ 1 + 2 Sum[ (1 + Mod[k, 2]) q^k / (1 - q^k + q^(2 k)), {k, n}], {q, 0, n}];
a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, q]^2 EllipticTheta[ 4, 0, q^3]^2 / (EllipticTheta[ 4, 0, q^2] EllipticTheta[ 4, 0, q^6]), {q, 0, n}];
a[ n_] := If[ n < 1, Boole[n == 0], 2 (-1)^n Sum[(-1)^(n/d) {2, -1, 0, 1, -2, 0}[[ Mod[ d, 6, 1] ]], {d, Divisors @ n}]];

PARI

{a(n) = if( n<1, n==0, 2 * sumdiv(n, d, (n/d%2 + 1) * (-1)^(d\3) * (d%3>0) ))};

PARI

{a(n) = if( n<1, n==0, 2 * (-1)^n * sumdiv(n, d, (-1)^(n/d) * [0, 2, -1, 0, 1, -2][d%6 + 1]))};

PARI

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^8 * eta(x^3 + A)^4 * eta(x^12 + A) / (eta(x + A)^4 * eta(x^4 + A)^3 * eta(x^6 + A)^4), n))};

PARI

{a(n) = my(A, p, e); if( n<1, n==0, A = factor(n); 4 * prod( k=1, matsize(A)[1], if( p=A[k, 1], e=A[k, 2]; if( p==2, 3/2*(e%2), if( p==3, 1, if( p%6 == 1, e+1, 1-e%2))))))};

Formula

Expansion of phi(q)^2 * phi(-q^3)^2 / (phi(-q^2) * phi(-q^6)) = psi(q) * psi(-q^3) * (chi(q) * chi(-q^3))^3 in powers of q where phi(), psi(), chi() are Ramanujan theta functions.

Expansion of (2*a(q) + 3*a(q^2) - 2*a(q^4)) / 3 = (b(q) - 2*b(q^4)) * (b(q) - 4*b(q^4)) / (3*b(q^2)) in powers of q where a(), b() are cubic AGM theta functions.

Expansion of eta(q^2)^8 * eta(q^3)^4 * eta(q^12) / (eta(q)^4 * eta(q^4)^3 * eta(q^6)^4) in powers of q.

Euler transform of period 12 sequence [ 4, -4, 0, -1, 4, -4, 4, -1, 0, -4, 4, -2, ...].

Moebius transform is period 12 sequence [ 4, 2, 0, -6, -4, 0, 4, 6, 0, -2, -4, 0, ...].

G.f. is a period 1 Fourier series which satisfies f(-1 / (12 t)) = 48^(1/2) (t/i) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A253625.

a(n) = 4*b(n) where b() is multiplicative with b(2^e) = (3/4) * (1 - (-1)^e) if e>0, b(3^e) = 1, b(p^e) = e+1 if p == 1 (mod 6), b(p^e) = (1 + (-1)^e)/2 if p == 5 (mod 6).

G.f.: 1 + Sum_{k>0} (1 + (k mod 2)) * q^k / (1 - q^k + q^(2*k)).

G.f.: Product_{k>0} (1 + q^k) * (1 - q^(2*k)) * (1 - q^(3*k)) * (1 + q^(6*k)) / ((1 + q^(2*k)) * (1 - q^k + q^(2*k)))^3.

a(n) = (-1)^n * A244339(n). a(2*n) = A004016(n). a(2*n + 1) = 4 * A033762(n). a(3*n) = a(n). a(6*n + 1) = 4 * A097195(n). a(6*n + 2) = 6 * A033687(n). a(6*n + 4) = a(6*n = 5) = 0.

a(12*n + 1) = 4 * A123884(n). a(12*n + 2) = 6 * A097195(n). a(12*n + 3) = 4 * A112604(n). a(12*n + 7) = 8 * A121361(n). a(12*n + 9) = 4 * A112605(n).

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 2*Pi/sqrt(3) = 3.627598... (A186706). - Amiram Eldar, Dec 30 2023