A258998 a(n) = -(-1)^n if n = k^2 for positive integer k, otherwise 0.

0, 1, 0, 0, -1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0

Offset

0

Comments

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

Denoted by B_n^o(-1) in Kassel and Reutenauer 2105.

Links

Seiichi Manyama, Table of n, a(n) for n = 0..1000

Christian Kassel and Christophe Reutenauer, On the Zeta Functions of Punctual Hilbert schemes of a Two-Dimensional Torus, arXiv:1505.07229 [math.AG], 2015, see page 31 7.2(d).

Michael Somos, Introduction to Ramanujan theta functions.

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions.

Formula

Expansion of (1 - phi(-x)) / 2 in powers of x where phi() is a Ramanujan theta function.

a(n) is multiplicative with a(2^e) = (-1)^(e/2) if e even, a(p^e) = 1 if p>2 and e even, otherwise 0.

G.f.: (1 - theta_4(x)) / 2 where theta_4() is a Jacobi theta function.

-2 * a(n) = A002448(n) unless n = 0. a(n) = -(-1)^n * A010052(n) unless n = 0.

a(n) = ((-1)^(n+2^(n-(floor(sqrt(n)))^2))-(-1)^(n+2^n))/2. - Luce ETIENNE, Aug 31 2015

Dirichlet g.f.: zeta(2*s)*(4^s-1)/(4^s+1). - Amiram Eldar, Dec 29 2022

Example

G.f. = x - x^4 + x^9 - x^16 + x^25 - x^36 + x^49 - x^64 + x^81 - x^100 + ...

Mathematica

a[ n_] := If[ n < 1, 0, If[ IntegerQ[ Sqrt @ n], -(-1)^n, 0]];
a[ n_] := SeriesCoefficient[ (1 - EllipticTheta[ 4, 0, x]) / 2, {x, 0, n}];

Prog

(PARI) {a(n) = if( n<1, 0, issquare(n), -(-1)^n, 0)};
(Magma) [((-1)^(n+2^(n-(Floor(Sqrt(n)))^2))-(-1)^(n+2^n))/2: n in [0..100]]; // Vincenzo Librandi, Sep 03 2015

Crossrefs

Cf. A002448, A010052.

Cf. A000007.

Cf. Related to Expansion of (1 - eta(q)^k / eta(q^k)) / k in powers of q: this sequence (k=2), A123477 (k=3), A109091 (k=5), A160535 (k=7).

Sequence in context: A258021 A033684 A080885 * A363713 A068716 A188020

Adjacent sequences: A258995 A258996 A258997 * A258999 A259000 A259001

Keyword

sign,mult

Author

Michael Somos, Jun 16 2015

Status

approved