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A096727
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Expansion of eta(q)^8/eta(q^2)^4 in powers of q.
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5
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1, -8, 24, -32, 24, -48, 96, -64, 24, -104, 144, -96, 96, -112, 192, -192, 24, -144, 312, -160, 144, -256, 288, -192, 96, -248, 336, -320, 192, -240, 576, -256, 24, -384, 432, -384, 312, -304, 480, -448, 144, -336, 768, -352, 288, -624, 576, -384, 96, -456, 744, -576, 336, -432, 960, -576, 192
(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).
Euler transform of period 2 sequence [ -8,-4,...].
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LINKS
| M. Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
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FORMULA
| G.f. Prod_{k>0} (1-x^k)^8/(1-x^(2k))^4 = 1 +Sum_{k>0} k(-8x^k/(1-x^k) +48x^(2k)/(1-x^(2k))-64x^(4k)/(1-x^(4k))).
G.f. theta_4(q)^4 = (Sum_{k} (-q)^(k^2))^4.
Expansion of phi(-q)^4 in powers of q where phi() is a Ramanujan theta function. - Michael Somos Nov 01 2006
G.f. A(x) satisfies 0=f(A(x), A(x^3), A(x^9)) where f(u, v, w) = v^4 -30*u*v^2*w +12*u*v*w*(u +9*w) -u*w*(u^2 +9*w*u +81*w^2).
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MATHEMATICA
| CoefficientList[ Series[1 + Sum[k(-8x^k/(1 - x^k) + 48x^(2k)/(1 - x^(2k)) - 64x^(4k)/(1 - x^(4k))), {k, 1, 60}], {x, 0, 60}], x] (from Robert G. Wilson v Jul 14 2004)
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PROG
| (PARI) a(n)=if(n<1, n==0, 8*(-1)^n*sumdiv(n, d, if(d%4, d)))
(PARI) a(n)=local(A); if(n<0, 0, A=x*O(x^n); polcoeff(eta(x+A)^8/eta(x^2+A)^4, n))
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CROSSREFS
| A000118(n)=(-1)^n*a(n). A109506(n)=a(n)/8 if n>0. A004011(n)=a(2n). A005879(n)=-a(2n+1).
Sequence in context: A162829 A175368 A000118 * A028660 A028644 A056196
Adjacent sequences: A096724 A096725 A096726 * A096728 A096729 A096730
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KEYWORD
| sign
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AUTHOR
| Michael Somos, Jul 06 2004
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