A112206 Coefficients of replicable function number "72b".

1, 1, 0, 2, 2, 1, 2, 2, 3, 4, 4, 4, 7, 7, 6, 10, 11, 11, 14, 16, 17, 21, 22, 24, 32, 34, 34, 44, 49, 50, 60, 66, 72, 84, 90, 98, 117, 125, 132, 156, 171, 181, 206, 226, 245, 277, 298, 322, 369, 397, 422, 480, 522, 557, 620, 674, 728, 807, 868, 936, 1043, 1121, 1198

Offset

0,4

Comments

From Michael Somos, Oct 28 2019: (Start)

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

Convolution squared is A112173.

G.f. is a period 1 Fourier series which satisfies f(-1 / (12 t)) = f(t) where q = exp(2 Pi i t).

Given G.f. A(x), then B(q) = q^(-1) * A(q^6) satisfies 0 = f(B(q), B(q^2), B(q^4)) where f(u, v, w) = 2 + (u^2 - v)*v*w^2 + (u^2 + v)*v^2.

(End)

Links

G. C. Greubel, Table of n, a(n) for n = 0..1000

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

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Index entries for McKay-Thompson series for Monster simple group

Formula

a(n) ~ exp(sqrt(2*n)*Pi/3) / (2^(5/4) * sqrt(3) * n^(3/4)). - Vaclav Kotesovec, Sep 08 2015

Expansion of q^(1/6)*((eta(q^2)*eta(q^6))^2/(eta(q)*eta(q^3)*eta(q^4) *eta(q^12))) in powers of q. - G. C. Greubel, Jun 01 2018

From Michael Somos, Oct 28 2019: (Start)

Expansion of chi(x) * chi(x^3) in powers of x where chi() is a Ramanujan theta function.

Euler transform of period 12 sequence [1, -1, 2, 0, 1, -2, 1, 0, 2, -1, 1, 0, ...].

G.f.: Product_{k>=0} (1 + x^(2*k + 1)) * (1 + x^(6*k + 3)).

a(n) = (-1)^n * A112175(n). a(2*n) = A328789(n). a(2*n + 1) = A328790(n).

(End)

Empirical: Sum_{n>=0} a(n) / exp(n*Pi) = (1/2) * exp(-Pi/6) * sqrt(2) * 3^(1/12) * Gamma(2/3)^(2/3) * Gamma(3/4)^(2/3) * (sqrt(2) * (1+3^(1/2)))^(2/3) / Gamma(11/12)^(2/3) / Pi^(1/3) = A388523. - Simon Plouffe, Sep 17 2025

Example

G.f. = 1 + x + 2*x^3 + 2*x^4 + x^5 + 2*x^6 + 2*x^7 + 3*x^8 + ...
G.f. = q^-1 + q^5 + 2*q^17 + 2*q^23 + q^29 + 2*q^35 + 2*q^41 + ...

Mathematica

nmax = 60; CoefficientList[Series[Product[(1 + x^k)*(1 + x^(3*k)) / ((1 + x^(2*k))*(1 + x^(6*k))), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Sep 08 2015 *)
eta[q_]:= q^(1/24)*QPochhammer[q]; h:= q^(1/6)*((eta[q^2]*eta[q^6])^2/(eta[q]*eta[q^3]*eta[q^4]*eta[q^12])); a:= CoefficientList[Series [h, {q, 0, 60}], q]; Table[a[[n]], {n, 1, 50}] (* G. C. Greubel, Jun 01 2018 *)
a[ n_] := SeriesCoefficient[ QPochhammer[ -x, x^2] QPochhammer[ -x^3, x^6], {x, 0 , n}]; (* Michael Somos, Oct 28 2019 *)

Prog

(PARI) q='q+O('q^50); h=((eta(q^2)*eta(q^6))^2/(eta(q)*eta(q^3)*eta(q^4) *eta(q^12))); Vec(h) \\ G. C. Greubel, Jun 01 2018
(PARI) {a(n) = my(A); if( n < 0, 0, A = x * O(x^n); polcoeff( (eta(x^2 + A) * eta(x^6 + A))^2 / (eta(x + A) * eta(x^3 + A) * eta(x^4 + A) * eta(x^12 + A)), n))}; /* Michael Somos, Oct 28 2019 */

Crossrefs

Cf. A112173, A112175, A328789, A328790.

Sequence in context: A097266 A226983 A112175 * A038541 A070215 A071457

Adjacent sequences: A112203 A112204 A112205 * A112207 A112208 A112209

Keyword

nonn

Author

Michael Somos, Aug 28 2005

Status

approved