A058490 Coefficients of replicable function number 12b.

1, 5, 27, 41, 146, 243, 510, 887, 1755, 2728, 5052, 7857, 13157, 20253, 32805, 48680, 76568, 112320, 169814, 246263, 365013, 519046, 755632, 1063368, 1516404, 2112551, 2972160, 4089098, 5683166, 7750782, 10633276, 14382932, 19539387, 26192432, 35263852

Offset: 0

Author

N. J. A. Sloane, Nov 27 2000

Keywords

• nonn

Comments

The convolution square of this sequence is A007254 except for the constant term: T12b(q)^2 = T6A(q^2) + 10. - G. A. Edgar, Apr 15 2017

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

Examples

			T12b = 1/q + 5*q + 27*q^3 + 41*q^5 + 146*q^7 + 243*q^9 + 510*q^11 + ...
		

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 0..10000 (terms 0..502 from G. A. Edgar)
• D. Alexander, C. Cummins, J. McKay and C. Simons, Completely Replicable Functions, LMS Lecture Notes, 165, ed. Liebeck and Saxl (1992), 87-98, annotated and scanned copy.
• D. Ford, J. McKay and S. P. Norton, More on replicable functions, Commun. Algebra 22, No. 13, 5175-5193 (1994).
• Michael Somos, Emails to N. J. A. Sloane, 1993
• 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

Crossrefs

Cf. A000521, A007240, A007254, A014708, A007241, A007267, A045478, etc.

Cf. A058484, A058492, A156215, A288630.

Programs

Mathematica

a[ n_] := With[{A = (QPochhammer[ x^2] QPochhammer[ x^3] / (QPochhammer[ x] QPochhammer[ x^6]))^6}, SeriesCoefficient[ A - x / A, {x, 0, n}]]; (* Michael Somos, Jun 12 2017 *)

PARI

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); A = (eta(x^2 + A) * eta(x^3 + A) / (eta(x + A) * eta(x^6 + A)))^6; polcoeff( A - x/A, n))}; /* Michael Somos, Jun 12 2017 */

PARI

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); A = (eta(x + A) * eta(x^3 + A) / (eta(x^2 + A) * eta(x^6 + A)))^3; polcoeff( A + 8*x/A, n))}; /* Michael Somos, Jun 12 2017 */

Formula

Expansion of q^(1/2) * (eta(q)^3*eta(q^3)^3 / (eta(q^2)^3*eta(q^6)^3) + 8 *eta(q^2)^3*eta(q^6)^3 / (eta(q)^3*eta(q^3)^3)) in powers of q. - G. A. Edgar, Apr 15 2017

From Michael Somos, Jun 12 2017: (Start)

Expansion of (chi(-x) * chi(-x^3))^3 + 8*x/(chi(-x) * chi(-x^3))^3 = (chi(-x^3) / chi(-x))^6 - x*(chi(-x) / chi(-x^3))^6 in powers of x.

G.f. is a period 1 Fourier series which satisfies f(-1 / (24 t)) = f(t) where q = exp(2 Pi i t).

Convolution square is A288630.

a(n) = 2*A058484(n) - A058206(n) = 2*A058492(n) - A058489(n). (End)

a(n) ~ exp(2*Pi*sqrt(n/3)) / (2 * 3^(1/4) * n^(3/4)). - Vaclav Kotesovec, Jun 13 2017

Extensions

More terms from Michael Somos, Feb 06 2009