A129134 Expansion of (1 - phi(-q) * phi(-q^2)) / 2 in powers of q where phi() is a Ramanujan theta function.

1, 1, -2, -1, 0, 2, 0, -1, 3, 0, -2, -2, 0, 0, 0, -1, 2, 3, -2, 0, 0, 2, 0, -2, 1, 0, -4, 0, 0, 0, 0, -1, 4, 2, 0, -3, 0, 2, 0, 0, 2, 0, -2, -2, 0, 0, 0, -2, 1, 1, -4, 0, 0, 4, 0, 0, 4, 0, -2, 0, 0, 0, 0, -1, 0, 4, -2, -2, 0, 0, 0, -3, 2, 0, -2, -2, 0, 0, 0, 0, 5, 2, -2, 0, 0, 2, 0, -2, 2, 0, 0, 0, 0, 0, 0, -2, 2, 1, -6, -1, 0, 4, 0, 0, 0

Offset

1,3

Comments

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

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

Links

G. C. Greubel, Table of n, a(n) for n = 1..5000

Michael Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Formula

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

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

a(n) = A002325(n) * (-1)^floor((n-1)/2}. A082564(n) = -2 * a(n) unless n=0.

a(3*n + 1) = A258747(n). a(3*n + 2) = A258764(n). - Michael Somos, Jun 09 2015

Example

G.f. = q + q^2 - 2*q^3 - q^4 + 2*q^6 - q^8 + 3*q^9 - 2*q^11 - 2*q^12 - q^16 + ...

Mathematica

a[ n_] := If[ n < 1, 0, (-1)^Quotient[ n - 1, 2] DivisorSum[n, KroneckerSymbol[-2, #] &]]; (* Michael Somos, Jun 09 2015 *)
a[ n_] := SeriesCoefficient[ (1 - EllipticTheta[ 4, 0, q] EllipticTheta[ 4, 0, q^2]) / 2 , {q, 0, n}]; (* Michael Somos, Jun 09 2015 *)
a[ n_] := SeriesCoefficient[ (1 - QPochhammer[ q]^2 QPochhammer[ q^2] / QPochhammer[ q^4]) / 2 , {q, 0, n}]; (* Michael Somos, Jun 09 2015 *)

Prog

(PARI) {a(n) = if( n<1, 0, (-1)^((n-1)\2) * sumdiv(n, d, kronecker( -2, d)))};
(PARI) {a(n) = local(A); if( n<1, 0, A = x * O(x^n); polcoeff( (1 - eta(x + A)^2 * eta(x^2 + A) / eta(x^4 + A)) / 2, n))};

Crossrefs

Cf. A002325, A002479, A082564, A258747, A258764.

Sequence in context: A322454 A133693 A002325 * A065675 A348128 A328777

Adjacent sequences: A129131 A129132 A129133 * A129135 A129136 A129137

Keyword

sign

Author

Michael Somos, Mar 30 2007

Status

approved