A112222 Coefficients of replicable function number "126a".

1, 0, 1, 1, 0, 0, 1, 1, 1, 1, 1, 1, 2, 1, 2, 2, 2, 2, 3, 2, 3, 4, 3, 4, 5, 4, 5, 6, 6, 6, 8, 7, 8, 9, 9, 10, 12, 11, 13, 14, 14, 15, 19, 17, 20, 22, 21, 23, 27, 26, 29, 32, 32, 34, 39, 38, 43, 46, 47, 50, 56, 55, 61, 67, 67, 72, 80, 79, 86, 93, 96, 101, 112, 112, 121, 130, 133

Offset

0,13

Comments

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

Links

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000

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

Michael Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Index entries for McKay-Thompson series for Monster simple group

Formula

Expansion of (psi(x^4) * phi(x^42) - x^10 * phi(x^22) * psi(x^84)) / (f(-x^6) * f(-x^14)) = (psi(x^12) * phi(x^14) - x^2 * phi(x^6) * psi(x^28)) / (f(-x^2) * f(-x^42)) in powers of x where phi(), psi(), f() are Ramanujan theta functions [Zhu Cao, May 24 2018]. - Michael Somos, May 24 2018

Convolution square is A112204, cube is A058675, sixth power is A058563.

a(n) ~ exp(2*Pi*sqrt(2*n/7)/3) / (2^(3/4) * sqrt(3) * 7^(1/4) * n^(3/4)). - Vaclav Kotesovec, May 30 2018

From G. C. Greubel, Jul 03 2018: (Start)

Expansion of sqrt(T63a) in powers of q, where T63a = A112204.

Expansion of (T21A)^(1/6) in powers of q, where T21A = A058563. (End)

Example

G.f. = 1 + x^2 + x^3 + x^6 + x^7 + x^8 + x^9 + x^10 + x^11 + 2*x^12 + x^13 + ...
T126a = 1/q + q^11 + q^17 + q^35 + q^41 + q^47 + q^53 + q^59 + q^65 +...

Mathematica

a[ n_] := With[ {A = (QPochhammer[ x^3] QPochhammer[ x^7] / (QPochhammer[ x] QPochhammer[ x^21]))^2}, SeriesCoefficient[ (A - 2 x + x^2/A)^(1/6), {x, 0, n}]]; (* Michael Somos, May 24 2018 *)
a[ n_] := With[ {A = QPochhammer[ x] QPochhammer[ x^3] / (QPochhammer[ x^7] QPochhammer[ x^21])}, SeriesCoefficient[ (A + x + 7 x^2/A)^(1/6), {x, 0, n}]]; (* Michael Somos, May 24 2018 *)
a[ n_] := SeriesCoefficient[ (1/2) x^(-3/4) (EllipticTheta[ 2, 0, x^3] EllipticTheta[ 3, 0, x^7] - EllipticTheta[ 3, 0, x^3] EllipticTheta[ 2, 0, x^7]) / (QPochhammer[ x] QPochhammer[ x^21]), {x, 0, n}, Assumptions -> x > 0]; (* Michael Somos, May 24 2018 *)
a[ n_] := SeriesCoefficient[ (1/2) x^(-1/4) (EllipticTheta[ 2, 0, x^1] EllipticTheta[ 3, 0, x^21] - EllipticTheta[ 3, 0, x^1] EllipticTheta[ 2, 0, x^21]) /(QPochhammer[ x^3] QPochhammer[ x^7]), {x, 0, n}, Assumptions -> x > 0]; (* Michael Somos, May 24 2018 *)
CoefficientList[Series[((QPochhammer[x^3]^2 * QPochhammer[x^7]^2 - x*QPochhammer[x]^2 * QPochhammer[x^21]^2) / (QPochhammer[x] * QPochhammer[x^3] * QPochhammer[x^7] * QPochhammer[x^21]))^(1/3), {x, 0, 100}], x] (* Vaclav Kotesovec, May 30 2018 *)
eta[q_] := q^(1/24)*QPochhammer[q]; nmax = 120; b:= eta[q]*eta[q^3]/ (eta[q^7]*eta[q^21]);  T21A := 1 + b + 7/b; T126a := CoefficientList[ Series[(q*T21A + O[q]^nmax)^(1/6), {q, 0, 100}], q]; Table[T126a[[n]], {n, 1, 80}] (* G. C. Greubel, Jul 03 2018 *)

Prog

(PARI) q='q+O('q^80); b = eta(q)*eta(q^3)/(q*eta(q^7)*eta(q^21)); T21A = b + 1 + 7/b; Vec((q*T21A)^(1/6)) \\ G. C. Greubel, Jul 03 2018

Crossrefs

Cf. A058563, A058675, A112204.

Sequence in context: A370809 A025832 A320385 * A112220 A185278 A241065

Adjacent sequences: A112219 A112220 A112221 * A112223 A112224 A112225

Keyword

nonn

Author

Michael Somos, Aug 28 2005

Status

approved