A176143 McKay-Thompson series of class 16C for the Monster group with a(0) = 2.

1, 2, 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

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

Formula

Expansion of (1/q) * chi(-q^4)^5 * chi(-q^8)^2 / (chi(-q)^2 * chi(-q^2)^5) = (1/q) * chi(q)^2 * chi(q^4)^2 * chi(-q^4)^7 / chi(-q^2)^7 = (1/q) * chi(-q^8)^7 / (chi(q)^5 * chi(-q)^7 * chi(q^4)^5) in powers of q where chi() is a Ramanujan theta function.

Expansion of eta(q^4)^10 / (eta(q)^2 * eta(q^2)^3 * eta(q^8)^3 * eta(q^16)^2) in powers of q.

Euler transform of period 16 sequence [2, 5, 2, -5, 2, 5, 2, -2, 2, 5, 2, -5, 2, 5, 2, 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) = A214035(n) unless n=0.

a(n) ~ exp(sqrt(n)*Pi) / (2^(3/2) * n^(3/4)). - Vaclav Kotesovec, May 01 2017

Expansion of (1/q) * chi(q)^2 * chi(q^2)^7 * chi(q^4)^2 in powers of q. - Michael Somos, Feb 09 2019

Example

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

Mathematica

QP = QPochhammer; s = QP[q^4]^10 / (QP[q]^2 * QP[q^2]^3 * QP[q^8]^3 * QP[q^16]^2) + O[q]^40; CoefficientList[s, q] (* Jean-François Alcover, Nov 14 2015, adapted from PARI *)
a[ n_] := SeriesCoefficient[ q^-1 QPochhammer[ -q, q^2]^2 QPochhammer[ -q^2, q^4]^7 QPochhammer[ -q^4, q^8]^2, {q, 0, n}]; (* Michael Somos, Feb 09 2019 *)

Prog

(PARI) {a(n) = my(A); if( n<-1, 0, n++; A = x * O(x^n); polcoeff( eta(x^4 + A)^10 / (eta(x + A)^2 * eta(x^2 + A)^3 * eta(x^8 + A)^3 * eta(x^16 + A)^2), n))};

Crossrefs

Cf. A058516, A214035.

Sequence in context: A212318 A346461 A232392 * A296946 A096227 A191309

Adjacent sequences: A176140 A176141 A176142 * A176144 A176145 A176146

Keyword

nonn

Author

Michael Somos, Aug 08 2012

Status

approved