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A123861
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Expansion of (f(q)*f(q^3)/(f(-q)*f(-q^3)))^2 in powers of q where f() is a Ramanujan theta function.
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1
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1, 4, 8, 20, 48, 88, 168, 320, 544, 932, 1584, 2544, 4080, 6488, 9984, 15288, 23232, 34568, 51144, 75152, 108832, 156736, 224352, 317728, 447648, 627292, 871856, 1206068, 1660416, 2271032, 3092976, 4194464, 5657728, 7602096, 10175760
(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).
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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
| Euler transform of period 12 sequence [ 4, -2, 8, 0, 4, -4, 4, 0, 8, -2, 4, 0, ...].
G.f. A(x) satisfies 0=f(A(x), A(x^2)) where f(u, v)=(u-1)^2 - 4*u*v*(v-1).
Let g.f. A(x)=u, then B(x)=(u-1)/4*u, B(x^2)=((u-1)/4)^2/u where B(x)=g.f. A123653.
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PROG
| (PARI) {a(n)=local(A); if(n<0, 0, A=x*O(x^n); polcoeff( (eta(x^2+A)*eta(x^6+A))^6/ (eta(x+A)*eta(x^3+A))^4/ (eta(x^4+A)*eta(x^12+A))^2, n))}
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CROSSREFS
| A123647(n)=a(n)/4 if n>0.
Sequence in context: A190589 A009916 A203167 * A115099 A060919 A009333
Adjacent sequences: A123858 A123859 A123860 * A123862 A123863 A123864
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KEYWORD
| nonn
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AUTHOR
| Michael Somos, Oct 14 2006
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