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A101195
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Expansion of q^(-1/2)theta_2(q^3)/theta_2(q) in powers of q^2.
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1
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1, -1, 1, -1, 2, -3, 3, -4, 6, -7, 8, -10, 13, -16, 18, -22, 28, -33, 38, -45, 55, -65, 74, -87, 104, -121, 138, -160, 188, -217, 247, -284, 330, -378, 428, -489, 562, -640, 722, -820, 936, -1059, 1191, -1345, 1524, -1717, 1924, -2163, 2438, -2734, 3054, -3419, 3834, -4284, 4770, -5321, 5943
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,5
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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).
Expansion of solution to a functional equation.
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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.: Product_{k>0} (1+x^k+x^(2k))(1-x^k+x^(2k))^2.
Expansion of q^(-1/4)eta(q)eta(q^6)^2/(eta(q^2)^2eta(q^3)) in powers of q.
Euler transform of period 6 sequence [ -1, 1, 0, 1, -1, 0, ...].
Given g.f. A(x), B(x)=A(x^4)x satisfies 0=f(B(x), B(x^3)) where f(u, v)=u^3-v+3uv^2-3u^2v^3.
Given g.f. A(x), B(x)=(A(x^4)x)^2 satisfies 0=f(B(x), B(x^2)) where f(u, v)=u^2-v+v^2+3u^2v.
Expansion of psi(q^3)/psi(q) in powers of q where psi() is a Ramanujan theta function.
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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^6+A)^2/ eta(x^2+A)^2/eta(x^3+A), n))
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CROSSREFS
| a(n)=(-1)^n*A036018(n).
Sequence in context: A041003 A067592 A036018 * A123552 A071610 A198726
Adjacent sequences: A101192 A101193 A101194 * A101196 A101197 A101198
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KEYWORD
| sign
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AUTHOR
| Michael Somos, Dec 03 2004
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