A131946 Expansion of (phi(-q) * phi(-q^3))^2 in powers of q where phi() is a Ramanujan theta function.

1, -4, 4, -4, 20, -24, 4, -32, 52, -4, 24, -48, 20, -56, 32, -24, 116, -72, 4, -80, 120, -32, 48, -96, 52, -124, 56, -4, 160, -120, 24, -128, 244, -48, 72, -192, 20, -152, 80, -56, 312, -168, 32, -176, 240, -24, 96, -192, 116, -228, 124, -72, 280, -216, 4

Offset

0,2

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).

References

N. J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; p. 85, Eq. (32.66).

Links

G. C. Greubel, Table of n, a(n) for n = 0..10000

Michael Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Formula

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

Expansion of (b(q)^2 / b(q^2)) * (c(q)^2 / c(q^2)) / 3 in powers of q where b(), c() are cubic AGM theta functions.

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

Euler transform of period 6 sequence [-4, -2, -8, -2, -4, -4, ...].

a(n) = -4 * b(n) where b() is multiplicative with b(2^e) = 3 - 2^(e+1), b(3^e) = 1, b(p^e) = (p^(e+1) - 1) / (p - 1) if p>3. - Michael Somos, Sep 19 2013

G.f. is a period 1 Fourier series which satisfies f(-1 / (6 t)) = 48 (t/i)^2 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A111932. - Michael Somos, Sep 19 2013

G.f.: 1 - 4 * (Sum_{k>0} k * (-x)^k / (1 - x^k) * Kronecker(9, k)) = (theta_3(-x) * theta_3(-x^3))^2.

a(n) = (-1)^n * A034896(n). a(n) = -4 * A131947(n) unless n = 0.

a(3*n) = a(n). a(2*n) = A125514(n). - Michael Somos, Sep 19 2013

Example

G.f. = 1 - 4*q + 4*q^2 - 4*q^3 + 20*q^4 - 24*q^5 + 4*q^6 - 32*q^7 + 52*q^8 + ...

Mathematica

a[ n_] := SeriesCoefficient[ (EllipticTheta[ 4, 0, q] EllipticTheta[ 4, 0, q^3])^2, {q, 0, n}]; (* Michael Somos, Sep 19 2013 *)
a[ n_] := SeriesCoefficient[ (QPochhammer[ q] QPochhammer[ q^3])^4 / (QPochhammer[ q^2] QPochhammer[ q^6])^2, {q, 0, n}]; (* Michael Somos, Sep 19 2013 *)
a[ n_] := If[ n < 1, Boole[n == 0], -4 Sum[ d {0, 1, -1, 0, -1, 1}[[ Mod[ d, 6] + 1]], {d, Divisors @ n}]]; (* Michael Somos, Sep 19 2013 *)
a[ n_] := If[ n < 1, Boole[n == 0], -4 Sum[ n/d {6, 1, -3, -2, -3, 1}[[ Mod[ d, 6] + 1]], {d, Divisors @ n}]]; (* Michael Somos, Sep 19 2013 *)

Prog

(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x + A) * eta(x^3 + A))^4 / (eta(x^2 + A) * eta(x^6 + A))^2, n))};
(PARI) {a(n) = if( n<1, n==0, -4 * sumdiv( n, d, d * [0, 1, -1, 0, -1, 1][d%6 + 1]))}; /* Michael Somos, Sep 19 2013 */
(Sage) A = ModularForms( Gamma0(6), 2, prec=55) . basis();  A[0] - 4*A[1] + 4*A[2]; # Michael Somos, Sep 19 2013
(Magma) A := Basis( ModularForms( Gamma0(6), 2), 55); A[1] - 4*A[2] + 4*A[3]; /* Michael Somos, Nov 11 2015 */

Crossrefs

Cf. A034896, A111932, A125514, A131947.

Sequence in context: A275858 A241094 A319257 * A034896 A320970 A216871

Adjacent sequences: A131943 A131944 A131945 * A131947 A131948 A131949

Keyword

sign

Author

Michael Somos, Jul 30 2007

Status

approved