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

Author

Michael Somos, Aug 28 2005

Keywords

• nonn

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)

Examples

			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 + ...
		

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

Crossrefs

Cf. A112173, A112175, A328789, A328790.

Programs

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 *)

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 */

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)