A214035 McKay-Thompson series of class 16C for the Monster group with a(0) = 4.

1, 4, 8, 16, 34, 64, 112, 192, 319, 512, 808, 1248, 1886, 2816, 4144, 6016, 8643, 12288, 17296, 24144, 33442, 45952, 62720, 85056, 114620, 153600, 204728, 271456, 358204, 470528, 615344, 801408, 1039621, 1343488, 1729920, 2219808, 2838920, 3619136, 4599664

Offset

-1,2

Comments

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

A058516, A176143, A214035, A215346 are all essentially the same sequence. - N. J. A. Sloane, Aug 08 2012

References

D. Ford, J. McKay and S. P. Norton, More on replicable functions, Commun. Algebra 22, No. 13, 5175-5193 (1994). See Table 4 16C.

Links

Seiichi Manyama, Table of n, a(n) for n = -1..10000

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

Expansion of (1/q) * phi(q) * phi(q^4) / (phi(-q) * psi(q^8)) in powers of q where phi(), psi() are Ramanujan theta functions.

Expansion of (eta(q^2) * eta(q^8))^6 / (eta(q) * eta(q^4) * eta(q^16))^4 in powers of q.

Euler transform of period 16 sequence [ 4, -2, 4, 2, 4, -2, 4, -4, 4, -2, 4, 2, 4, -2, 4, 0, ...].

G.f. is a period 1 Fourier series which satisfies f(-1 / (16 t)) = f(t) where q = exp(2 Pi i t).

a(n) = A058516(n) = A176143(n) unless n=0. a(n) = -(-1)^n * A215346(n). Convolution square of A058630. Convolution inverse of A215349.

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

Example

G.f. = 1/q + 4 + 8*q + 16*q^2 + 34*q^3 + 64*q^4 + 112*q^5 + 192*q^6 + 319*q^7 + ...

Mathematica

a[ n_] := SeriesCoefficient[ q^-1 QPochhammer[ -q, q^2]^4 QPochhammer[ -q^2, q^4]^2 QPochhammer[ -q^4, q^8]^4, {q, 0, n}] (* Michael Somos, Oct 25 2013 *)
nmax = 50; CoefficientList[Series[Product[((1-x^(2*k)) * (1-x^(8*k)))^6 / ((1-x^k) * (1-x^(4*k)) * (1-x^(16*k)))^4, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Sep 10 2015 *)

Prog

(PARI) {a(n) = local(A); if( n<-1, 0, n++; 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)^2, n))}

Crossrefs

Cf. A058516, A058630, A176143, A215346, A215349.

Sequence in context: A264166 A330913 A215346 * A040014 A141069 A306621

Adjacent sequences: A214032 A214033 A214034 * A214036 A214037 A214038

Keyword

nonn

Author

Michael Somos, Aug 08 2012

Status

approved