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A007267
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Expansion of 16 * (1 + k^2)^4 /(k * k'^2)^2 in powers of q where k is the Jacobian elliptic modulus, k' the complementary modulus and q is the nome.
(Formerly M5369)
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199
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1, 104, 4372, 96256, 1240002, 10698752, 74428120, 431529984, 2206741887, 10117578752, 42616961892, 166564106240, 611800208702, 2125795885056, 7040425608760, 22327393665024, 68134255043715, 200740384538624
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OFFSET
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-1,2
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COMMENTS
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McKay-Thompson series of class 2A for the Monster group with a(0) = 104.
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REFERENCES
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J. M. Borwein and P. B. Borwein, Pi and the AGM, Wiley, 1987, p. 195.
R. Fricke, Die elliptischen Funktionen und ihre Anwendungen, Teubner, 1922, Vol. 2, see p. 517.
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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FORMULA
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Expansion of 16 * (1 + k'^2)^4 /(k' * k^2)^2 in powers of q^2. - Michael Somos, Nov 11 2006
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EXAMPLE
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G.f. = 1/q + 104 + 4372*q + 96256*q^2 + 1240002*q^3 + 10698752*q^4 + ...
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MATHEMATICA
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a[ n_] := If[ n < -1, 0, With[ {m = InverseEllipticNomeQ[ q]}, SeriesCoefficient[ 16 (1 + m)^4 /(m (1 - m)^2), {q, 0, n}]]]; (* Michael Somos, Jun 29 2011 *)
a[ n_] := If[ n < -1, 0, With[ {m = ModularLambda[ Log[q]/(Pi I)]}, SeriesCoefficient[ 16 (1 + m)^4 /(m (1 - m)^2), {q, 0, n}]]]; (* Michael Somos, Jun 30 2011 *)
QP = QPochhammer; A = (QP[q]/QP[q^2])^12; s = (A + 64*(q/A))^2 + O[q]^30; CoefficientList[s, q] (* Jean-François Alcover, Nov 16 2015, adapted from PARI *)
nmax = 20; CoefficientList[Series[128*x + Product[1/(1 + x^k)^24, {k, 1, nmax}] + 4096*x^2*Product[(1 + x^k)^24, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Jun 03 2018 *)
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PROG
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(PARI) {a(n) = my(A); if( n<-1, 0, A = prod(k=1, n\2 + 1, 1 - x^(2*k - 1), 1 + x^2 * O(x^n))^12; polcoeff( (64 * x / A + A)^2, n+1))};
(PARI) {a(n) = my(A); if( n<-1, 0, n++; A = x * O(x^n); A = (eta(x + A) / eta(x^2 + A))^12; polcoeff( (A + 64 * x / A)^2, n))}; /* Michael Somos, Nov 11 2006 */
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CROSSREFS
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KEYWORD
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nonn,nice
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AUTHOR
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STATUS
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approved
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