A007250 McKay-Thompson series of class 4a for the Monster group.

1, -76, -702, -5224, -23425, -98172, -336450, -1094152, -3188349, -8913752, -23247294, -58610304, -140786308, -328793172, -740736900, -1629664840, -3486187003, -7307990208, -14976155896, -30157221352, -59594117256, -115975615160, -222119374922, -419704427016

Offset: 0

Author

N. J. A. Sloane

Keywords

• sign
• easy

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

Examples

			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 - ...
		

References

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

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000 (terms 0..499 from G. A. Edgar)
• 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).
• Masao Koike, Modular forms on non-compact arithmetic triangle groups, Unpublished manuscript [Extensively annotated with OEIS A-numbers by N. J. A. Sloane, Feb 14 2021. I wrote 2005 on the first page but the internal evidence suggests 1997.]
• 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

Crossrefs

Cf. A007242, A007249, A007260, A022577.

Programs

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 *)

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

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