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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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2
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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
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OFFSET
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0,2
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COMMENTS
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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
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REFERENCES
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N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
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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.
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FORMULA
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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
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
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EXAMPLE
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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 - ...
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MAPLE
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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) );
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MATHEMATICA
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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 *)
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PROG
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(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
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CROSSREFS
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KEYWORD
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sign,easy
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AUTHOR
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STATUS
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approved
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