A030216 Expansion of q^-1 * eta(q^10) * eta(q^14) in powers of q^2.

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

Offset

0,36

Comments

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

Links

Seiichi Manyama, Table of n, a(n) for n = 0..10000

M. Koike, On McKay's conjecture, Nagoya Math. J., 95 (1984), 85-89.

Michael Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Formula

G.f.: Product_{k>=1} (1 - x^(5*k)) * (1 - x^(7*k)). - Seiichi Manyama, Oct 18 2016

Expansion of f(-x^5) * f(-x^7) in powers of x where f() is a Ramanujan theta function.

Euler transform of period 35 sequence [ 0, 0, 0, 0, -1, 0, -1, 0, 0, -1, 0, 0, 0, -1, -1, 0, 0, 0, 0, -1, -1, 0, 0, 0, -1, 0, 0, -1, 0, -1, 0, 0, 0, 0, -2, ...]. - Michael Somos, Oct 19 2016

Example

G.f. = 1 - x^5 - x^7 - x^10 + x^12 - x^14 + x^17 + x^19 + x^24 + x^25 - x^32 + ...
G.f. = q - q^11 - q^15 - q^21 + q^25 - q^29 + q^35 + q^39 + q^49 + q^51 - q^65 + ...

Mathematica

a[ n_] := SeriesCoefficient[ QPochhammer[ x^5] QPochhammer[ x^7], {x, 0, n}]; (* Michael Somos, Oct 21 2016 *)

Prog

(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^5 + A) * eta(x^7 + A), n))}; /* Michael Somos, Oct 19 2016 */

Crossrefs

Cf. Expansion of eta(q^k)*eta(q^(24 - k)): A030199 (k=1), A030201 (k=3), A030213 (k=5), A030214 (k=7), A030215 (k=9), this sequence (k=10), A030217 (k=11).

Cf. A277582.

Sequence in context: A025438 A220400 A285108 * A357352 A339376 A362425

Adjacent sequences: A030213 A030214 A030215 * A030217 A030218 A030219

Keyword

sign

Author

N. J. A. Sloane

Status

approved