A132130 McKay-Thompson series of class 10D for the Monster group with a(0) = 6.

1, 6, 21, 62, 162, 378, 819, 1680, 3276, 6138, 11145, 19662, 33840, 57048, 94362, 153432, 245757, 388218, 605466, 933414, 1423614, 2149586, 3215844, 4769544, 7016572, 10243896, 14848809, 21378276, 30582360, 43484304, 61473438, 86428896

Offset

-1,2

Comments

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

The g.f. is denoted by x_10 in Cooper 2012.

Links

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

S. Cooper, Sporadic sequences, modular forms and new series for 1/pi, Ramanujan J. (2012).

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 q^(-1) * (chi(-q^5) / chi(-q))^6 in powers of q where chi() is a Ramanujan theta function.

Expansion of (eta(q^2) * eta(q^5) / (eta(q) * eta(q^10)))^6 in powers of q.

Euler transform of period 10 sequence [ 6, 0, 6, 0, 0, 0, 6, 0, 6, 0, ...].

G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^4)) where f(u, v, w) = (v - u^2) * (v - w^2) - u*w * (12*(1 + v^2) - 20*v).

G.f. is a period 1 Fourier series which satisfies f(-1 / (10 t)) = f(t) where q = exp(2 Pi i t).

G.f.: x^(-1) * (Product_{k>0} (1 + x^k) / (1 + x^(5*k)))^6.

G.f.: 1 / ( x * Product_{k>0} P(10,x^k)^6 ) where P(n,x) is the n-th cyclotomic polynomial.

a(n) = A058100(n) unless n=0.

a(n) ~ exp(2*Pi*sqrt(2*n/5)) / (2^(3/4) * 5^(1/4) * n^(3/4)). - Vaclav Kotesovec, Sep 06 2015

Example

G.f. = 1/q + 6 + 21*q + 62*q^2 + 162*q^3 + 378*q^4 + 819*q^5 + 1680*q^6 + ...

Mathematica

a[ n_] := SeriesCoefficient[ q^-1 (QPochhammer[ q^5, q^10] / QPochhammer[ q, q^2])^6, {q, 0, n}]; (* Michael Somos, Dec 07 2013 *)

Prog

(PARI) {a(n) = my(A); if( n<-1, 0, n++; A = x * O(x^n); polcoeff( (eta(x^2 + A) * eta(x^5 + A) / (eta(x + A) * eta(x^10 + A)))^6, n))};

Crossrefs

Cf. A058100.

Sequence in context: A048476 A122678 A256569 * A022571 A321947 A291226

Adjacent sequences: A132127 A132128 A132129 * A132131 A132132 A132133

Keyword

nonn,changed

Author

Michael Somos, Aug 11 2007, Aug 09 2008

Extensions

Edited by N. J. A. Sloane, May 16 2008 at the suggestion of R. J. Mathar

Status

approved