login
This site is supported by donations to The OEIS Foundation.

 

Logo

Thanks to everyone who made a donation during our annual appeal!
To see the list of donors, or make a donation, see the OEIS Foundation home page.

Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A132130 McKay-Thompson series of class 10D for the Monster group with a(0) = 6. 4
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 (list; graph; refs; listen; history; text; internal format)
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).

M. 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

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

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified January 20 02:13 EST 2019. Contains 319320 sequences. (Running on oeis4.)