A058632 Coefficients of replicable function number "32b".

1, 2, 3, 6, 7, 10, 16, 20, 29, 40, 52, 70, 91, 116, 149, 190, 242, 306, 383, 478, 590, 730, 897, 1096, 1342, 1630, 1975, 2390, 2873, 3448, 4133, 4932, 5880, 6994, 8290, 9814, 11587, 13650, 16058, 18848, 22089, 25842, 30178, 35186, 40950, 47594, 55231, 63996, 74068, 85592, 98776, 113864

Offset

-1,2

Comments

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

Links

G. C. Greubel, Table of n, a(n) for n = -1..1000

D. Ford, J. McKay and S. P. Norton, More on replicable functions, Comm. Algebra 22, No. 13, 5175-5193 (1994).

Index entries for McKay-Thompson series for Monster simple group

Formula

Expansion of q^(1/4)*(eta(q^2)*eta(q^4)/(eta(q)*eta(q^8)))^2 in powers of q. -G. C. Greubel, Jun 23 2018

a(n) ~ exp(sqrt(n/2)*Pi) / (2^(7/4) * n^(3/4)). - Vaclav Kotesovec, Jun 28 2018

From Michael Somos, Jan 17 2023: (Start)

Expansion of (psi(x)/psi(-x^2))^2 = (phi(-x^4)/psi(-x))^2 = (chi(x)*chi(x^2))^2 = (chi(-x^4)/chi(-x))^2 in powers of x where phi(), psi(), chi() are Ramanujan theta functions.

G.f. is a period 1 Fourier series which satisfies f(-1 / (128 t)) = f(t) where q = exp(2 Pi i t). (End)

Example

T32b = 1/q + 2*q^3 + 3*q^7 + 6*q^11 + 7*q^15 + 10*q^19 + 16*q^23 + 20*q^27 + ...

Mathematica

eta[q_] := q^(1/24)*QPochhammer[q]; a := CoefficientList[Series[q^(1/4)*(eta[q^2]*eta[q^4]/(eta[q]*eta[q^8]))^2, {q, 0, 60}], q]; Table[a[[n]], {n, 0, 50}] (* G. C. Greubel, Jun 23 2018 *)
From Michael Somos, Jan 17 2023: (Start)
a[ n_] := SeriesCoefficient[ (EllipticTheta[2, 0, q^(1/2)]/EllipticTheta[2, Pi/4, q])^2 / (2/q^(1/4)), {q, 0, n}];
a[ n_] := SeriesCoefficient[ (EllipticTheta[3, 0, -q^4]/EllipticTheta[2, Pi/4, q^(1/2)])^2 *(2*q^(1/4)), {q, 0, n}];
a[ n_] := SeriesCoefficient[ (QPochhammer[-q, q^2]*QPochhammer[-q^2, q^4])^2, {q, 0, n}];
a[ n_] := SeriesCoefficient[ (QPochhammer[+q^4, q^8]/QPochhammer[+q, q^2])^2, {q, 0, n}]; (End)

Prog

(PARI) q='q+O('q^50); Vec((eta(q^2)*eta(q^4)/(eta(q)*eta(q^8)))^2) \\ G. C. Greubel, Jun 23 2018

Crossrefs

Cf. A000521, A007240, A014708, A007241, A007267, A045478, etc.

Sequence in context: A288672 A086495 A051733 * A086583 A107852 A362286

Adjacent sequences: A058629 A058630 A058631 * A058633 A058634 A058635

Keyword

nonn

Author

N. J. A. Sloane, Nov 27 2000

Extensions

Terms a(6) onward added by G. C. Greubel, Jun 23 2018

Status

approved