A107635 McKay-Thompson series of class 32a for the Monster group.

1, 3, 3, 4, 9, 12, 15, 21, 30, 43, 54, 69, 94, 123, 153, 193, 252, 318, 391, 486, 609, 754, 918, 1119, 1376, 1680, 2019, 2432, 2946, 3540, 4220, 5034, 6015, 7157, 8463, 9999, 11835, 13956, 16374, 19206, 22542, 26376, 30750, 35829, 41745, 48526, 56250

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

Expansion of chi(x)^3 = phi(x) / psi(-x) in powers of x where phi(), psi(), chi() are Ramanujan theta functions.

Given g.f. A(x), then B(q) = A(q^8) / q satisfies 0 = f(B(q), B(q^3)) where f(u, v) = (u^3 - v) * (v^3 - u) - 9*u*v.

Euler transform of period 4 sequence [3, -3, 3, 0, ...].

G.f.: Product_{k>0} (1 + (-x)^k)^-3.

a(n) = (-1)^n * A022598(n).

a(n) ~ exp(Pi*sqrt(n/2)) / (2^(7/4) * n^(3/4)). - Vaclav Kotesovec, Aug 27 2015

G.f.: exp(3*Sum_{k>=1} x^k/(k*(1 - (-x)^k))). - Ilya Gutkovskiy, Jun 07 2018

Example

G.f. = 1 + 3*x + 3*x^2 + 4*x^3 + 9*x^4 + 12*x^5 + 15*x^6 + 21*x^7 + ...
T32a = 1/q + 3*q^7 + 3*q^15 + 4*q^23 + 9*q^31 + 12*q^39 + 15*q^47 + ...

Mathematica

a[ n_] := SeriesCoefficient[ (QPochhammer[ x^2]^2 / (QPochhammer[ x] QPochhammer[ x^4]))^3, {x, 0, n}]; (* Michael Somos, Jun 29 2014 *)
nmax = 50; CoefficientList[Series[Product[(1 + x^(2*k+1))^3, {k, 0, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Aug 27 2015 *)

Prog

(PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x^2 + A)^2 / (eta(x + A) * eta(x^4 + A)))^3, n))};

Crossrefs

Cf. A022598.

Sequence in context: A045794 A065678 A022598 * A132319 A130626 A175796

Adjacent sequences: A107632 A107633 A107634 * A107636 A107637 A107638

Keyword

nonn

Author

Michael Somos, May 18 2005

Status

approved