A030201 -id:A030201 - cp's OEIS frontend

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

N. J. A. Sloane

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

			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 + ...

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

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.

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

{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 */

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

A030213 Expansion of eta(q^5)*eta(q^19).

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane

sign

Seiichi Manyama, Table of n, a(n) for n = 0..10000
M. Koike, On McKay's conjecture, Nagoya Math. J., 95 (1984), 85-89.

Cf. Expansion of eta(q^k)*eta(q^(24 - k)): A030199 (k=1), A030201 (k=3), this sequence (k=5), A030214 (k=7), A030215 (k=9), A030216 (k=10), A030217 (k=11).

eta = QPochhammer;
CoefficientList[q eta[q^5] eta[q^19] + O[q]^100, q] (* Jean-François Alcover, Feb 21 2021 *)

Expansion of x * Product_{k>=1} (1 - x^(5*k)) * (1 - x^(19*k)). - Seiichi Manyama, Oct 18 2016

A030214 Expansion of eta(q^7)*eta(q^17).

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane

sign

Seiichi Manyama, Table of n, a(n) for n = 0..10000
M. Koike, On McKay's conjecture, Nagoya Math. J., 95 (1984), 85-89.

Cf. Expansion of eta(q^k)*eta(q^(24 - k)): A030199 (k=1), A030201 (k=3), A030213 (k=5), this sequence (k=7), A030215 (k=9), A030216 (k=10), A030217 (k=11).

eta = QPochhammer;
CoefficientList[q eta[q^7] eta[q^17] + O[q]^100, q] (* Jean-François Alcover, Feb 21 2021 *)

q='q+O('q^99); concat(0, Vec(eta(q^7)*eta(q^17))) \\ Charles R Greathouse IV, Oct 19 2016

Expansion of x * Product_{k>=1} (1 - x^(7*k)) * (1 - x^(17*k)). - Seiichi Manyama, Oct 19 2016

A030215 Expansion of eta(q^9)*eta(q^15).

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane

sign

Seiichi Manyama, Table of n, a(n) for n = 0..10000
M. Koike, On McKay's conjecture, Nagoya Math. J., 95 (1984), 85-89.

Cf. Expansion of eta(q^k)*eta(q^(24 - k)): A030199 (k=1), A030201 (k=3), A030213 (k=5), A030214 (k=7), this sequence (k=9), A030216 (k=10), A030217 (k=11).

eta = QPochhammer;
CoefficientList[q eta[q^9] eta[q^15] + O[q]^100, q] (* Jean-François Alcover, Feb 21 2021 *)

Expansion of x * Product_{k>=1} (1 - x^(9*k)) * (1 - x^(15*k)). - Seiichi Manyama, Oct 18 2016

A030217 Expansion of eta(q^11)*eta(q^13).

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane

sign

Seiichi Manyama, Table of n, a(n) for n = 0..10000
M. Koike, On McKay's conjecture, Nagoya Math. J., 95 (1984), 85-89.

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), A030216 (k=10), this sequence (k=11).

eta = QPochhammer;
CoefficientList[q eta[q^11] eta[q^13] + O[q]^100, q] (* Jean-François Alcover, Feb 21 2021 *)

Expansion of x * Product_{k>=1} (1 - x^(11*k)) * (1 - x^(13*k)). - Seiichi Manyama, Oct 18 2016

cp's OEIS Frontend

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

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A030213 Expansion of eta(q^5)*eta(q^19).

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A030214 Expansion of eta(q^7)*eta(q^17).

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A030215 Expansion of eta(q^9)*eta(q^15).

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A030217 Expansion of eta(q^11)*eta(q^13).

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