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A007250
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McKay-Thompson series of class 4a for the Monster group.
(Formerly M5353)
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0
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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; internal format)
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OFFSET
| 0,2
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REFERENCES
| 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, preprint.
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.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
| Index entries for McKay-Thompson series for Monster simple group
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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).
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EXAMPLE
| 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 - ...
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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
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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 *)
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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( A - 64 * x / A, n))} /* Michael Somos, Jul 22 2011 */
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CROSSREFS
| Cf. A007249, A022577.
Sequence in context: A200167 A178262 A185481 * A137061 A136963 A136964
Adjacent sequences: A007247 A007248 A007249 * A007251 A007252 A007253
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KEYWORD
| sign,easy
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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