A128516 McKay-Thompson series of class 14C for the Monster group with a(0) = 4.

1, 4, 10, 24, 51, 100, 190, 340, 585, 984, 1606, 2564, 4022, 6188, 9382, 14044, 20746, 30308, 43836, 62784, 89153, 125588, 175542, 243656, 335988, 460388, 627178, 849676, 1145024, 1535416, 2049200, 2722544, 3601681, 4745208, 6227276, 8141656

Offset

-1,2

Comments

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

Links

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

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^7) / chi(-q))^4 in powers of q where chi() is a Ramanujan theta function.

Expansion of (eta(q^2) * eta(q^7) / (eta(q) * eta(q^14)))^4 in powers of q.

Euler transform of period 14 sequence [ 4, 0, 4, 0, 4, 0, 0, 0, 4, 0, 4, 0, 4, 0, ...].

G.f. is a period 1 Fourier series which satisfies f(-1 / (14 t)) = f(t) where q = exp(2 Pi i t).

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 * (8*(1 + v^2) - 16*v).

G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = (u*v + 1) * (u + v) * (u - v)^2 - u*v * (u - 1) * (v - 1) * (u*v - 8*(u+v) + 1).

G.f.: (1/x) * (Product_{k>0} P(x^k))^-4 where P(x) = x^6 - x^5 + x^4 - x^3 + x^2 - x + 1 is the 14th cyclotomic polynomial.

a(n) = A058504(n) unless n = 0.

a(n) ~ exp(2*Pi*sqrt(2*n/7)) / (2^(3/4) * 7^(1/4) * n^(3/4)). - Vaclav Kotesovec, Oct 14 2015

Example

G.f. = 1/q + 4 + 10*q + 24*q^2 + 51*q^3 + 100*q^4 + 190*q^5 + 340*q^6 + ...

Mathematica

a[ n_] := SeriesCoefficient[ 1/q (QPochhammer[ -q, q] / QPochhammer[ -q^7, q^7])^4, {q, 0, n}]; (* Michael Somos, Apr 26 2015 *)
a[ n_] := SeriesCoefficient[ 1/q (QPochhammer[ q^2] QPochhammer[ q^7] / (QPochhammer[ q] QPochhammer[ q^14]))^4, {q, 0, n}]; (* Michael Somos, Apr 26 2015 *)
nmax=60; CoefficientList[Series[Product[((1+x^k) / (1+x^(7*k)))^4, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Oct 14 2015 *)

Prog

(PARI) {a(n) = my(A); if( n<-1, 0, n++; A = x * O(x^n); polcoeff( (eta(x^2 + A) * eta(x^7 + A) / (eta(x + A) * eta(x^14 + A)))^4, n))};

Crossrefs

Cf. A058504.

Sequence in context: A209970 A211392 A309777 * A022569 A093831 A274582

Adjacent sequences: A128513 A128514 A128515 * A128517 A128518 A128519

Keyword

nonn

Author

Michael Somos, Mar 06 2007

Status

approved