A058630 McKay-Thompson series of class 32B for the Monster group.

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

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).

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 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