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A007250 McKay-Thompson series of class 4a for the Monster group.
(Formerly M5353)
2
1, -76, -702, -5224, -23425, -98172, -336450, -1094152, -3188349, -8913752, -23247294, -58610304, -140786308, -328793172, -740736900, -1629664840, -3486187003, -7307990208, -14976155896, -30157221352, -59594117256, -115975615160, -222119374922, -419704427016 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

A more correct name would be: Expansion of replicable function of class 4a. See Alexander et al., 1992. - N. J. A. Sloane, Jun 12 2015

REFERENCES

Masao Koike, Modular forms on non-compact arithmetic triangle groups, preprint.

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

LINKS

G. A. Edgar, Table of n, a(n) for n = 0..499

D. Alexander, C. Cummins, J. McKay and C. Simons, Completely replicable functions, LMS Lecture Notes, 165, ed. Liebeck and Saxl (1992), 87-98, annotated and scanned copy.

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.

Index entries for McKay-Thompson series for Monster simple group

FORMULA

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, Jul 22 2011

a(n) = A007249(n) - 64 * A022577(n-1).

Expansion of q^(1/2) * ((eta(q) / eta(q^2))^12 - 64*(eta(q^2) / eta(q))^12) in powers of q. - G. A. Edgar, Mar 10 2017

EXAMPLE

G.f. = 1 - 76*x - 702*x^2 - 5224*x^3 - 23425*x^4 - 98172*x^5 - 336450*x^6 + ...

T4a = 1/q - 76*q - 702*q^3 - 5224*q^5 - 23425*q^7 - 98172*q^9 - ...

MAPLE

A022577L := proc(n)

        mul((1+x^m)^12, m=1..n+1) ;

        taylor(%, x=0, n+1) ;

        gfun[seriestolist](%) ;

end proc:

A007249L := proc(n)

        if n = 0 then

                0 ;

        else

                mul(1/(1+x^m)^12, m=1..n+1) ;

                taylor(%, x=0, n+1) ;

                gfun[seriestolist](%) ;

        end if;

end proc:

a022577 := A022577L(80) ;

a007249 := A007249L(80) ;

printf("1, ");

for i from 1 to 78 do

        printf("%d, ", op(i+1, a007249)-64*op(i, a022577) );

end do: # R. J. Mathar, Sep 30 2011

MATHEMATICA

a[ n_] := Module[ {m = InverseEllipticNomeQ @ q, e}, e = (1 - m) / (m / 16)^(1/2); SeriesCoefficient[ (e - 64 / e) q^(1/2), {q, 0, n}]]; (* Michael Somos, Jul 22 2011 *)

QP = QPochhammer; A = (QP[q]/QP[q^2])^12; s = A - 64*(q/A) + O[q]^30; CoefficientList[s, q] (* Jean-François Alcover, Nov 15 2015, adapted from PARI *)

nmax = 30; CoefficientList[Series[Product[((1-x^k) / (1-x^(2*k)))^12, {k, 1, nmax}] - 64*x*Product[((1-x^(2*k)) / (1-x^k))^12, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Mar 11 2017 *)

PROG

(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); A = (eta(x + A) / eta(x^2 + A))^12; polcoeff( A - 64 * x / A, n))}; /* Michael Somos, Jul 22 2011 */

(PARI) N=66; q='q+O('q^N); t=(eta(q)/eta(q^2))^12; Vec(t - 64*q/t) \\ Joerg Arndt, Mar 11 2017

CROSSREFS

Cf. A007242, A007249, A007260, A022577.

Sequence in context: A178262 A253411 A185481 * A137061 A230944 A136963

Adjacent sequences:  A007247 A007248 A007249 * A007251 A007252 A007253

KEYWORD

sign,easy

AUTHOR

N. J. A. Sloane

STATUS

approved

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Last modified December 15 03:08 EST 2017. Contains 296020 sequences.