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A112222 Coefficients of replicable function number "126a". 1
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 (list; graph; refs; listen; history; text; internal format)
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).

M. 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: A289122 A025832 A320385 * A112220 A185278 A241065

Adjacent sequences:  A112219 A112220 A112221 * A112223 A112224 A112225

KEYWORD

nonn

AUTHOR

Michael Somos, Aug 28 2005

STATUS

approved

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Last modified January 16 06:59 EST 2019. Contains 319188 sequences. (Running on oeis4.)