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

Author

Michael Somos, May 18 2005

Keywords

• nonn

Comments

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

Examples

			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 + ...
		

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

Crossrefs

Cf. A022598.

Programs

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 *)

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))};

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