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A058490 Coefficients of replicable function number 12b. 4
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 (list; graph; refs; listen; history; text; internal format)
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).

M. Somos, Emails to N. J. A. Sloane, 1993

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 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 A136917 A181890

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

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Last modified January 21 19:08 EST 2019. Contains 319350 sequences. (Running on oeis4.)