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A132002
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Expansion of phi(q^3)/phi(q) in powers of q where phi() is a Ramanujan theta function.
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2
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1, -2, 4, -6, 10, -16, 24, -36, 52, -74, 104, -144, 198, -268, 360, -480, 634, -832, 1084, -1404, 1808, -2316, 2952, -3744, 4728, -5946, 7448, -9294, 11556, -14320, 17688, -21780, 26740, -32736, 39968, -48672, 59122, -71644, 86616, -104484, 125768, -151072, 181104, -216684
(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
| Expansion of eta(q)^2* eta(q^4)^2* eta(q^6)^5/( eta(q^2)^5* eta(q^3)^2* eta(q^12)^2) in powers of q.
Euler transform of period 12 sequence [ -2, 3, 0, 1, -2, 0, -2, 1, 0, 3, -2, 0, ...].
G.f. A(x) satisfies 0= f(A(x), A(x^2)) where f(u, v)= (v+u)* (v-u) +(1 -u*v)* (1 -3*u*v).
G.f. A(x) satisfies 0= f(A(x), A(x^3)) where f(u, v)= u^3 -v +3*u*v^2* (1-u*v).
G.f.: (Sum_k x^(3k^2))/(Sum_k x^k^2) = Product_{k>0} (1+x^k+x^(2k))* (1-x^k+x^(2k))^3/ (1-x^(2k)+x^(4k))^2.
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PROG
| (PARI) {a(n)= if(n<0, 0, polcoeff( sum(k=1, sqrtint(n\3), 2*x^(3*k^2), 1+x*O(x^n))/ sum(k=1, sqrtint(n), 2*x^k^2, 1+x*O(x^n)), n))}
(PARI) {a(n)= local(A); if(n<0, 0, A= x*O(x^n); polcoeff( eta(x+A)^2* eta(x^4+A)^2* eta(x^6+A)^5/ eta(x^2+A)^5/ eta(x^3+A)^2/ eta(x^12+A)^2, n))}
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CROSSREFS
| a(n)= (-1)^n* A098151(n).
Sequence in context: A132212 A137414 A098151 * A028445 A006305 A067247
Adjacent sequences: A131999 A132000 A132001 * A132003 A132004 A132005
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KEYWORD
| sign
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AUTHOR
| Michael Somos, Aug 06 2007
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