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

0,2

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).

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

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

Example

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

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 *)

Prog

(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 */

Crossrefs

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

Cf. A058484, A058492, A156215, A288630.

Sequence in context: A298441 A056154 A156215 * A299578 A308829 A136917

Adjacent sequences: A058487 A058488 A058489 * A058491 A058492 A058493

Keyword

nonn

Author

N. J. A. Sloane, Nov 27 2000

Extensions

More terms from Michael Somos, Feb 06 2009

Status

approved