A004530 Expansion of (theta_2(0, x) + theta_3(0, x) + theta_4(0, x)) / 2 in powers of x^(1/4).

1, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 2, 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, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 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, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1

Offset

0,17

References

J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 120.

Links

Antti Karttunen, Table of n, a(n) for n = 0..65537

Formula

G.f.: (theta_2(0, x^4) + theta_3(0, x^4) + theta_4(0, x^4)) / 2.

Example

G.f. = 1 + x + x^9 + 2*x^16 + x^25 + x^49 + 2*x^64 + x^81 + x^121 + 2*x^144 + x^169 + ...

Mathematica

a[ n_] := SeriesCoefficient[ (EllipticTheta[ 2, 0, x^4] + EllipticTheta[ 3, 0, x^4] + EllipticTheta[4, 0, x^4]) / 2, {x, 0, n}]; (* Michael Somos, Aug 28 2017 *)
a[ n_] := With[{m = Sqrt@n}, Which[ n <= 0, Boole[n == 0], IntegerQ[m] && (Mod[m, 4] != 2), 2 - Mod[m, 2], True, 0]]; (* Michael Somos, Aug 28 2017 *)

Prog

(PARI) {a(n) = my(m); if( n<=0, n==0, issquare(n, &m) && (m%4!=2), 2-m%2, 0)}; /* Michael Somos, Aug 28 2017 */
(PARI) {a(n) = my(A); if(n<0, 0, A = x * O(x^n); polcoeff( x * eta(x^16 + A)^2 / eta(x^8 + A) + (eta(x^8 + A)^5 / (eta(x^4 + A) * eta(x^16 + A))^2 + eta(x^4 + A)^2 / eta(x^8 + A)) / 2, n))}; /* Michael Somos, Aug 28 2017 */

Crossrefs

Sequence in context: A069849 A138045 A072507 * A368980 A355549 A347438

Adjacent sequences: A004527 A004528 A004529 * A004531 A004532 A004533

Keyword

nonn

Author

N. J. A. Sloane

Extensions

Data section extended up to a(121) by Antti Karttunen, Aug 28 2017

Status

approved