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A022599
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Expansion of Product (1+q^m)^(-4); m=1..inf.
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1
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1, -4, 6, -8, 17, -28, 38, -56, 84, -124, 172, -232, 325, -448, 594, -784, 1049, -1388, 1796, -2320, 3005, -3864, 4912, -6216, 7877, -9940, 12430, -15488, 19309, -23972, 29580, -36408, 44766, -54876, 66978, -81536, 99150, -120272, 145374, -175344, 211242
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,2
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COMMENTS
| Ramanujan theta functions: f(q) := Prod_{k>=1} (1-(-q)^k) (see A121373), phi(q) := theta_3(q) := Sum_{k=-oo..oo} q^(k^2) (A000122), psi(q) := Sum_{k=0..oo} q^(k*(k+1)/2) (A10054), chi(q) := Prod_{k>=0} (1+q^(2k+1)) (A000700).
McKay-Thompson series of class 12J for the Monster group.
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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).
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.
T. J. I'a. Bromwich, Introduction to the Theory of Infinite Series, Macmillan, 2nd. ed. 1949, p. 116, q_2^4.
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LINKS
| M. Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
Index entries for McKay-Thompson series for Monster simple group
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FORMULA
| Expansion of chi(-q)^4 in powers of q where chi() is a Ramanujan theta function.
Expansion of q^(1/6) * (eta(q) / eta(q^2))^4 in powers of q.
Euler transform of period 2 sequence [ -4, 0, ...]. - Michael Somos Apr 26 2008
Given G.f. A(x) then B(x) = (A(x^6) / x)^2 satisfies 0 = f(B(x), B(x^2)) where f(u, v) = u * (16 + u * v) - v^2. - Michael Somos Apr 26 2008
Given G.f. A(x) then B(x) = A(x^6) / x satisfies 0 = f(B(x), B(x^2), B(x^4)) where f(u, v, w) = 4 * v * (v + u^2) - w^2 * (v - u^2). - Michael Somos Apr 26 2008
G.f. is a period 1 Fourier series which satisfies f(-1/(72 t)) = 4 / f(t) where q = exp(2 pi i t).
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EXAMPLE
| T12J = 1/q - 4q^5 + 6q^11 - 8q^17 + 17q^23 - 28q^29 + 38q^35 + ...
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PROG
| (PARI) {a(n)=local(A); if(n<0, 0, A=x*O(x^n); polcoeff( (eta(x+A)/eta(x^2+A))^4, n))}
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CROSSREFS
| Convolution inverse of A022569.
Sequence in context: A083166 A185292 * A112160 A132040 A114315 A058238
Adjacent sequences: A022596 A022597 A022598 * A022600 A022601 A022602
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KEYWORD
| sign
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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