|
|
A129134
|
|
Expansion of (1 - phi(-q) * phi(-q^2)) / 2 in powers of q where phi() is a Ramanujan theta function.
|
|
4
|
|
|
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
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
1,3
|
|
COMMENTS
|
For n nonzero, a(n) is nonzero if and only if n is in A002479.
|
|
LINKS
|
|
|
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.
|
|
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
|
|
|
KEYWORD
|
sign
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
|