A133692 Expansion of phi(-q) * phi(q^2) in powers of q where phi() is a Ramanujan theta function.

1, -2, 2, -4, 2, 0, 4, 0, 2, -6, 0, -4, 4, 0, 0, 0, 2, -4, 6, -4, 0, 0, 4, 0, 4, -2, 0, -8, 0, 0, 0, 0, 2, -8, 4, 0, 6, 0, 4, 0, 0, -4, 0, -4, 4, 0, 0, 0, 4, -2, 2, -8, 0, 0, 8, 0, 0, -8, 0, -4, 0, 0, 0, 0, 2, 0, 8, -4, 4, 0, 0, 0, 6, -4, 0, -4, 4, 0, 0, 0, 0

Offset

0,2

Comments

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

a(n) is nonzero if and only if n is in A002479.

Links

G. C. Greubel, Table of n, a(n) for n = 0..5000

Michael Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Formula

Expansion of eta(q)^2 * eta(q^4)^5 / (eta(q^2)^3 * eta(q^8)^2) in powers of q.

Euler transform of period 8 sequence [ -2, 1, -2, -4, -2, 1, -2, -2, ...].

Moebius transform is period 16 sequence [ -2, 4, -2, 0, 2, 4, 2, 0, -2, -4, -2, 0, 2, -4, 2, 0, ...].

a(n) = -2 * b(n) where b(n) is multiplicative with b(2^e) = -1 if e>0, b(p^e) = (1 + (-1)^e) / 2 if p == 5, 7 (mod 8), b(p^e) = e + 1 if p == 1, 3 (mod 8).

G.f.: Product_{k>0} (1 - x^k)^2 * (1 + x^(2*k))^3 / (1 + x^(4*k))^2.

a(8*n + 5) = a(8*n + 7) = 0.

A133690 is the convolution square. a(n) = (-1)^n * A033715(n). a(2*n) = A033715(n). a(2*n + 1) = -2 * A113411(n).

Example

G.f. = 1 - 2*q + 2*q^2 - 4*q^3 + 2*q^4 + 4*q^6 + 2*q^8 - 6*q^9 - 4*q^11 + ...

Mathematica

a[ n_] := If[ n < 1, Boole[n == 0], (-1)^n 2 DivisorSum[ n, KroneckerSymbol[ -2, #] &]]; (* Michael Somos, Oct 30 2015 *)
a[ n_] := SeriesCoefficient[ EllipticTheta[ 4, 0, q] EllipticTheta[ 3, 0, q^2], {q, 0, n}]; (* Michael Somos, Aug 29 2014 *)

Prog

(PARI) {a(n) = if( n<1, n==0, (-1)^n * 2 * sumdiv(n, d, kronecker( -2, d)))};
(PARI) {a(n) = my(A); if ( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A)^2 * eta(x^4 + A)^5 / (eta(x^2 + A)^3 * eta(x^8 + A)^2), n))};
(Magma) A := Basis( ModularForms( Gamma1(16), 1), 80); A[1] -2*A[2] +2*A[3] - 4*A[4] + 2*A[5] + 4*A[7]; /* Michael Somos, Aug 29 2014 */

Crossrefs

Cf. A002479, A033715, A113441, A133690.

Sequence in context: A114427 A129355 A080963 * A033715 A082564 A139093

Adjacent sequences: A133689 A133690 A133691 * A133693 A133694 A133695

Keyword

sign

Author

Michael Somos, Sep 20 2007

Status

approved