A212771 McKay-Thompson series of class 24B for the Monster group with a(0) = 2.

1, 2, 3, 8, 11, 16, 31, 40, 58, 96, 125, 176, 262, 336, 457, 640, 819, 1088, 1464, 1864, 2420, 3168, 3991, 5088, 6533, 8160, 10267, 12976, 16061, 19968, 24912, 30576, 37648, 46464, 56616, 69136, 84518, 102288, 123961, 150304, 180805, 217664, 262042, 313472

Offset

-1,2

Links

G. C. Greubel, Table of n, a(n) for n = -1..1000

Formula

Expansion of (eta(q^2) * eta(q^4) * eta(q^6) * eta(q^12) / (eta(q) * eta(q^3) * eta(q^8) * eta(q^24)) )^2 in powers of q.

Euler transform of period 24 sequence [ 2, 0, 4, -2, 2, 0, 2, 0, 4, 0, 2, -4, 2, 0, 4, 0, 2, 0, 2, -2, 4, 0, 2, 0, ...].

Expansion of b(q^2) * c(q^2) * b(q^4) * c(q^4) / (b(q) * c(q) * b(q^8) * c(q^8)) in powers of q where b(), c() are cubic AGM theta functions.

G.f. is a period 1 Fourier series which satisfies f(-1 / (24 t)) = f(t) where q = exp(2 Pi i t).

a(n) = A058572(n) unless n=0. a(2*n) = 2 * A123861(n). Convolution inverse of A212770.

a(n) ~ exp(sqrt(2*n/3)*Pi) / (2^(5/4)*3^(1/4)*n^(3/4)). - Vaclav Kotesovec, Sep 07 2017

Example

1/q + 2 + 3*q + 8*q^2 + 11*q^3 + 16*q^4 + 31*q^5 + 40*q^6 + 58*q^7 + ...

Mathematica

QP = QPochhammer; s = (QP[q^2]*QP[q^4]*QP[q^6]*(QP[q^12]/(QP[q]*QP[q^3]* QP[q^8]*QP[q^24])))^2 + O[q]^50; CoefficientList[s, q] (* Jean-François Alcover, Nov 16 2015, adapted from PARI *)

Prog

(PARI) {a(n) = local(A); if( n<-1, 0, n = n+1; A = x * O(x^n); polcoeff( (eta(x^2 + A) * eta(x^4 + A) * eta(x^6 + A) * eta(x^12 + A) / (eta(x + A) * eta(x^3 + A) * eta(x^8 + A) * eta(x^24 + A)) )^2, n))}

Crossrefs

Cf. A058572, A123861, A212770.

Sequence in context: A050557 A039000 A059671 * A119116 A138756 A291202

Adjacent sequences: A212768 A212769 A212770 * A212772 A212773 A212774

Keyword

nonn

Author

Michael Somos, May 26 2012

Status

approved