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!)
A052241 McKay-Thompson series of class 8C for Monster. 3
1, 26, 79, 326, 755, 2106, 4460, 10284, 20165, 41640, 77352, 147902, 263019, 475516, 816065, 1413142, 2353446, 3936754, 6391091, 10390150, 16497734, 26184098, 40775677, 63394792, 97037170, 148178934, 223351867, 335704742, 499050461, 739575640, 1085723797 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

LINKS

G. C. Greubel, Table of n, a(n) for n = 0..1000

J. H. Conway and S. P. Norton, Monstrous Moonshine, Bull. Lond. Math. Soc. 11 (1979) 308-339.

D. Ford, J. McKay and S. P. Norton, More on replicable functions, Commun. Algebra 22, No. 13, 5175-5193 (1994).

J. McKay and H. Strauss, The q-series of monstrous moonshine and the decomposition of the head characters, Comm. Algebra 18 (1990), no. 1, 253-278.

M. Somos, Emails to N. J. A. Sloane, 1993

Index entries for McKay-Thompson series for Monster simple group

FORMULA

Expansion of 2 * q^(1/4) * ((k'^4 + 4*k^2) / (k'^2 * k))^(1/2) in powers of q. - Michael Somos, Sep 01 2014

Given g.f. A(x), then B(q) = A(x^q) / q satisfies 0 = f(B(q), B(q^3)) where f(u, v) = (u^2 + v^2)^2 - (u*v - 12) * (u*v - 32)^2. - Michael Somos, Sep 01 2014

Given g.f. A(x), then B(q) = A(x^q) / q satisfies 0 = f(B(q), B(q^3)) where f(u, v) = (u^2 - v^2)^2 - (u*v - 48) * (u*v - 16)^2. - Michael Somos, Sep 01 2014

G.f. is a period 1 Fourier series which satisfies f(-1 / (8 t)) = f(t) where q = exp(2 Pi i t). - Michael Somos, Sep 01 2014

Convolution square is A007247. Convolution fourth power is A007267.

a(n) ~ exp(Pi*sqrt(2*n)) / (2^(5/4)*n^(3/4)). - Vaclav Kotesovec, May 01 2017

EXAMPLE

G.f. = 1 + 26*x + 79*x^2 + 326*x^3 + 755*x^4 + 2106*x^5 + 4460*x^6 + ...

T8C = 1/q + 26*q^3 + 79*q^7 + 326*q^11 + 755*q^15 + 2106*q^19 + 4460*q^23 + ...

MATHEMATICA

QP = QPochhammer; A = O[q]^40; A = (QP[q + A]/QP[q^2 + A])^12; s = Sqrt[A + 64*(q/A)]; CoefficientList[s, q] (* Jean-Fran├žois Alcover, Nov 13 2015, adapted from PARI *)

eta[q_] := q^(1/24)*QPochhammer[q]; e4D := q^(1/2)*(eta[q]/eta[q^2])^12;

T4B := e4D + 64*q/e4D; a[n_]:= SeriesCoefficient[Sqrt[(T4B /. {q -> q^2}) + O[q]^100], {q, 0, n}]; Table[a[n], {n, 0, 50}][[1 ;; ;; 2]] (* G. C. Greubel, Feb 13 2018 *)

PROG

(PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); A = (eta(x + A) / eta(x^2 + A))^12; polcoeff( sqrt(A + 64 * x / A), n))}; /* Michael Somos, Sep 01 2014 */

CROSSREFS

Cf. A007247, A007249, A007267.

Sequence in context: A042326 A042328 A042330 * A042332 A260199 A239172

Adjacent sequences:  A052238 A052239 A052240 * A052242 A052243 A052244

KEYWORD

nonn

AUTHOR

N. J. A. Sloane, Nov 27 2000

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 17 18:42 EST 2019. Contains 319251 sequences. (Running on oeis4.)