A134461 Expansion of (phi(x) * psi(-x))^4 in powers of x where phi(), psi() are Ramanujan theta functions.

1, 4, -2, -24, -11, 44, 22, -8, 50, -44, -96, 56, -121, -152, 198, 160, 176, 48, -162, 88, -198, -52, 22, -528, 233, 200, -242, -88, -176, 668, 550, 264, -44, -188, 224, -728, 154, -484, -1056, 656, -311, -236, -100, 792, 714, -528, 640, 88, -478, -484, 1566, 968, 192, 780, -1994, -648, -942

Offset: 0

Author

Michael Somos, Oct 26 2007

Keywords

• sign

Comments

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

Number 34 of the 74 eta-quotients listed in Table I of Martin (1996).

Examples

			G.f. = 1 + 4*x - 2*x^2 - 24*x^3 - 11*x^4 + 44*x^5 + 22*x^6 - 8*x^7 + ...
G.f. = q + 4*q^3 - 2*q^5 - 24*q^7 - 11*q^9 + 44*q^11 + 22*q^13 - 8*q^15 + ...
		

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..10000
• Y. Martin, Multiplicative eta-quotients, Trans. Amer. Math. Soc. 348 (1996), no. 12, 4825-4856, see page 4852 Table I.
• Michael Somos, Index to Yves Martin's list of 74 multiplicative eta-quotients and their A-numbers
• Michael Somos, Introduction to Ramanujan theta functions
• Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Crossrefs

Cf. A216711.

The same as A030211 except for signs.

Programs

Magma

A := Basis( CuspForms( Gamma0(16), 4), 115); A[1] + 4*A[3]; /* Michael Somos, Jun 10 2015 */

Mathematica

a[ n_] := SeriesCoefficient[ (EllipticTheta[ 4, 0, x^2] EllipticTheta[ 2, 0, x^(1/2)] / (2 x^(1/8)))^4, {x, 0, n}]; (* Michael Somos, Jun 10 2015 *)

PARI

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

Formula

Expansion of q^(-1/2) * (eta(q^2)^4 / (eta(q) * eta(q^4)))^4 in powers of q.

Euler transform of period 4 sequence [ 4, -12, 4, -8, ...].

a(n) = b(2*n + 1) where b() is multiplicative with b(2^e) = 0^e, b(p^e) = b(p)*b(p^(e-1)) - p^3*b(p^(e-2)).

G.f. is a period 1 Fourier series which satisfies f(-1 / (16 t)) = 256 (t/i)^4 f(t) where q = exp(2 Pi i t).

G.f.: (Product_{k>0} (1 + x^k) * (1 - x^(2*k))^2 / (1 + x^(2*k)))^4.

a(n) = (-1)^n * A030211(n).

Convolution square is A216711. - Michael Somos, Jun 10 2015