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Search: id:A029841
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| A029841 |
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G.f.: ( Product_{k>0} (1+q^(2k-1))/(1+q^(2k)) )^4. |
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+0
3
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| 1, 4, 2, -8, -1, 20, -2, -40, 3, 72, 2, -128, -4, 220, -4, -360, 5, 576, 8, -904, -8, 1384, -10, -2088, 11, 3108, 12, -4552, -15, 6592, -18, -9448, 22, 13392, 26, -18816, -29, 26216, -34, -36224, 38, 49700, 42, -67728, -51, 91688
(list; graph; listen)
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OFFSET
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0,2
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COMMENT
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Hauptmodul for Gamma'_0(8).
McKay-Thompson series of class 8E for the Monster group.
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REFERENCES
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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 A. Sebbar, Fuchsian groups, automorphic functions and Schwarzians, Math. Ann., 318 (2000), 255-275.
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.
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LINKS
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T. D. Noe, Table of n, a(n) for n=0..1000
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FORMULA
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Euler transform of period 4 sequence [4,-8,4,0,...]. - Michael Somos Mar 18 2004
eta(4z)^12/(eta(2z)^4*eta(8z)^8) = 1/q + 4q + 2q^3 - 8q^5 - q^7 + 20q^9 - ... Also eta(2z)^4/eta(8z)^4.
Given g.f. A(x), then B(x)=A(x^2)/x satisfies 0=f(B(x), B(x^2)) where f(u, v)=16+8v+v^2-u^2*v - Michael Somos Mar 18 2004
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EXAMPLE
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T8E = 1/q +4*q +2*q^3 -8*q^5 -q^7 +20*q^9 -2*q^11 -40*q^13 +...
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PROGRAM
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(PARI) a(n)=if(n<0, 0, X=x+x*O(x^n); polcoeff((eta(X^2)^3/eta(X)/eta(X^4)^2)^4, n))
(PARI) a(n)=local(A, m); if(n<0, 0, A=1+O(x); m=1; while(m<=n, m*=2; A=subst(A, x, x^2); A=(4*x+A)/sqrt(A)); polcoeff(A, n))
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CROSSREFS
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A029839(n)=a(2n). A079006(n)=a(2n+1)/4. - Michael Somos Mar 27 2004.
Sequence in context: A050105 A128333 A019953 this_sequence A112143 A112151 A112152
Adjacent sequences: A029838 A029839 A029840 this_sequence A029842 A029843 A029844
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KEYWORD
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sign,easy,nice
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AUTHOR
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njas
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