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A058630 McKay-Thompson series of class 32B for the Monster group. 2
1, 2, 2, 4, 7, 10, 14, 20, 27, 36, 50, 64, 84, 110, 140, 180, 229, 288, 360, 452, 560, 692, 854, 1044, 1275, 1554, 1884, 2276, 2745, 3296, 3950, 4724, 5630, 6696, 7946, 9408, 11115, 13108, 15422, 18112, 21238, 24850, 29034, 33864, 39429, 45844, 53224, 61696 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

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

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, Comm. Algebra 22, No. 13, 5175-5193 (1994).

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 chi(x)^2 * chi(x^2) * chi(x^4)^2 in powers of x where chi() is a Ramanujan theta function. - Michael Somos, Oct 25 2013

Contribution from Michael Somos, Aug 08 2012 (Start)

Expansion of f(x) * f(x^4) / (f(-x) * f(-x^16)) = psi(x) * psi(x^4) / (psi(-x) * psi(x^8)) = chi(x) * chi(x^4) * chi(-x^8) / chi(-x) = (phi(x) * phi(x^4) / (phi(-x) * psi(x^8)))^(1/2) in powers of x where phi(), psi(), chi(), f() are Ramanujan theta functions.

Expansion of q^(1/2) * (eta(q^2) * eta(q^8))^3 / (eta(q) * eta(q^4) * eta(q^16))^2 in powers of q.

Euler transform of period 16 sequence [ 2, -1, 2, 1, 2, -1, 2, -2, 2, -1, 2, 1, 2, -1, 2, 0, ...].

G.f. is a period 1 Fourier series which satisfies f(-1 / (64 t)) = f(t) where q = exp(2 Pi i t).

Convolution square is A214035. (end)

Expansion of psi(-x^2)^2 / (phi(-x) * psi(x^8)) = phi(x) * phi(x^4) / psi(-x^2)^2 in powers of x. - Michael Somos, Dec 14 2014

a(n) ~ exp(Pi*sqrt(n/2)) / (2^(7/4) * n^(3/4)). - Vaclav Kotesovec, Sep 10 2015

EXAMPLE

G.f. = 1 + 2*x + 2*x^2 + 4*x^3 + 7*x^4 + 10*x^5 + 14*x^6 + 20*x^7 + 27*x^8 + ...

T32B = 1/q + 2*q + 2*q^3 + 4*q^5 + 7*q^7 + 10*q^9 + 14*q^11 + 20*q^13 + ...

MATHEMATICA

a[ n_] := SeriesCoefficient[ QPochhammer[ -x, x]^2 QPochhammer[ -x^4, x^4] / (QPochhammer[-x^2, x^2] QPochhammer[-x^8, x^8]^2), {x, 0, n}]; (* Michael Somos, Oct 25 2013 *)

a[ n_] := SeriesCoefficient[ QPochhammer[ -x, x^2]^2 QPochhammer[ -x^2, x^4] QPochhammer[ -x^4, x^8]^2, {x, 0, n}]; (* Michael Somos, Dec 14 2014 *)

nmax = 50; CoefficientList[Series[Product[((1-x^(2*k)) * (1-x^(8*k)))^3 / ((1-x^k) * (1-x^(4*k)) * (1-x^(16*k)))^2, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Sep 10 2015 *)

PROG

(PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x^2 + A) * eta(x^8 + A))^3 / (eta(x + A) * eta(x^4 + A) * eta(x^16 + A))^2, n))}; /* Michael Somos, Aug 08 2012 */

CROSSREFS

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

Cf. A214035.

Sequence in context: A134791 A095760 A082222 * A095092 A094686 A277752

Adjacent sequences:  A058627 A058628 A058629 * A058631 A058632 A058633

KEYWORD

nonn

AUTHOR

N. J. A. Sloane, Nov 27 2000

STATUS

approved

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Last modified October 14 14:06 EDT 2019. Contains 328017 sequences. (Running on oeis4.)