A058539 McKay-Thompson series of class 18d for the Monster group.

1, 4, 10, 20, 35, 60, 100, 164, 261, 400, 600, 884, 1291, 1864, 2656, 3740, 5205, 7184, 9842, 13388, 18082, 24244, 32300, 42784, 56378, 73928, 96466, 125284, 161981, 208568, 267524, 341880, 435343, 552424, 698666, 880848, 1107229, 1387804, 1734624, 2162248

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 (chi(-x^3) / chi(-x))^4 in powers of x where chi() is a Ramanujan theta function.

Expansion of q^(1/3) * c(q) * b(q^2) / (b(q) * c(q^2)) in powers of q where b(), c() are cubic AGM theta functions.

Expansion of q^(1/3) * (eta(q^2) * eta(q^3) / (eta(q) * eta(q^6)))^4 in powers of q.

Given g.f. A(x), then B(x) = A(x^3) / x satisfies 0 = f(B(x), B(x^2)) where f(u, v) = 8 * (u * v)^2 - (1 + u * v) * (u^2 - v) * (v^2 - u).

Given g.f. A(x), then B(x) = A(x^3) / x satisfies 0 = f(B(x), B(x^2)) where f(u, v) = 9 * (u * v)^2 - (u - v^2 + u^2*v) * (v - u^2 + u*v^2).

Given g.f. A(x), then B(x) = A(x^3) / x satisfies 0 = f(B(x), B(x^2), B(x^4)) where f(u, v, w) = 8 * u * v * w - (u^2 - v) * (w^2 - v).

Given g.f. A(x), then B(x) = A(x^3) / x satisfies 0 = f(B(x), B(x^5)) where f(u, v) = u*v * (1 + 25 * u*v + u^2*v^2)^2 - (u^3 + v^3 + 10 * u*v * (1 + u*v))^2.

G.f. is a period 1 Fourier series which satisfies f(-1 / (54 t)) = f(t) where q = exp(2 Pi i t).

Convolution square of A103262. Convolution fourth power of A003105.

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

Example

1 + 4*x + 10*x^2 + 20*x^3 + 35*x^4 + 60*x^5 + 100*x^6 + 164*x^7 + ...
T18d = 1/q + 4*q^2 + 10*q^5 + 20*q^8 + 35*q^11 + 60*q^14 + 100*q^17 + ...

Mathematica

nmax = 50; CoefficientList[Series[Product[((1+x^(3*k-1))*(1+x^(3*k-2)))^4, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Sep 10 2015 *)
QP = QPochhammer; s = (QP[q^2]*(QP[q^3]/(QP[q]*QP[q^6])))^4 + O[q]^40; CoefficientList[s, q] (* Jean-François Alcover, Nov 30 2015, adapted from PARI *)

Prog

(PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x^2 + A) * eta(x^3 + A) / (eta(x + A) * eta(x^6 + A)))^4, n))} /* Michael Somos, Mar 04 2012 */

Crossrefs

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

Cf. A003105, A103262.

Sequence in context: A137359 A134987 A261636 * A354001 A368174 A353999

Adjacent sequences: A058536 A058537 A058538 * A058540 A058541 A058542

Keyword

nonn

Author

N. J. A. Sloane, Nov 27 2000

Status

approved