A128517 McKay-Thompson series of class 18E for the Monster group with a(0) = 3.

1, 3, 6, 13, 24, 42, 73, 120, 192, 299, 456, 684, 1007, 1464, 2100, 2976, 4176, 5802, 7993, 10920, 14808, 19946, 26688, 35496, 46944, 61752, 80826, 105286, 136536, 176304, 226725, 290448, 370704, 471467, 597600, 755028, 950980, 1194216

Offset

-1,2

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 = -1..1000

Vaclav Kotesovec, A method of finding the asymptotics of q-series based on the convolution of generating functions, arXiv:1509.08708 [math.CO], Sep 30 2015

Michael Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Formula

Expansion of q^(-1) * (chi(-q^9) / chi(-q))^3 in powers of q where chi() is a Ramanujan theta function.

Expansion of (eta(q^2) * eta(q^9) / (eta(q) * eta(q^18)))^3 in powers of q.

Euler transform of period 18 sequence [ 3, 0, 3, 0, 3, 0, 3, 0, 0, 0, 3, 0, 3, 0, 3, 0, 3, 0, ...].

G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^4)) where f(u, v, w) = (u^2 - v) * (w^2 - v) - u*w * (6*(1 + v^2) - 10*v).

G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = (u*v - u - v)^3 - u*v* (u + v - 1)^3.

G.f. is a period 1 Fourier series which satisfies f(-1 / (18 t)) = f(t) where q = exp(2 Pi i t).

G.f.: (1/x) * (Product_{k>0} P(x^k))^-3 where P(x) = (x^2 - x + 1) * (x^6 - x^3 + 1).

a(n) = A058535(n) unless n = 0.

a(n) ~ exp(2*Pi*sqrt(2*n)/3) / (2^(3/4) * sqrt(3) * n^(3/4)). - Vaclav Kotesovec, Oct 13 2015

Example

G.f. = 1/q + 3 + 6*q + 13*q^2 + 24*q^3 + 42*q^4 + 73*q^5 + 120*q^6 + ...

Mathematica

a[ n_] := SeriesCoefficient[ 1/q (QPochhammer[ -q, q] / QPochhammer[ -q^9, q^9])^3, {q, 0, n}]; (* Michael Somos, Apr 26 2015 *)
a[ n_] := SeriesCoefficient[ 1/q (QPochhammer[ q^2] QPochhammer[ q^9] / (QPochhammer[ q] QPochhammer[ q^18]))^3, {q, 0, n}]; (* Michael Somos, Apr 26 2015 *)
nmax=60; CoefficientList[Series[Product[((1+x^k) / (1+x^(9*k)))^3, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Oct 13 2015 *)

Prog

(PARI) {a(n) = my(A); if( n<-1, 0, n++; A = x * O(x^n); polcoeff( (eta(x^2 + A) * eta(x^9 + A) / (eta(x + A) * eta(x^18 + A)))^3, n))};

Crossrefs

Cf. A058535.

Sequence in context: A058554 A342646 A342853 * A022568 A120006 A263847

Adjacent sequences: A128514 A128515 A128516 * A128518 A128519 A128520

Keyword

nonn

Author

Michael Somos, Mar 06 2007

Status

approved