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A128111
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Expansion of q^(-1)*(phi(q)/ phi(q^9) -1)/2 in powers of q^3 where phi() is a Ramanujan theta function.
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7
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1, 1, 0, -2, -2, 1, 4, 4, -1, -8, -8, 2, 14, 14, -4, -24, -23, 6, 40, 38, -10, -63, -60, 16, 98, 92, -24, -150, -140, 36, 224, 208, -54, -329, -304, 78, 478, 440, -112, -684, -627, 160, 968, 884, -224, -1358, -1236, 312, 1884, 1710, -432, -2592, -2346, 590, 3540, 3196, -801, -4796, -4320, 1082, 6454
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,4
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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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REFERENCES
| B. C. Berndt, Ramanujan's Notebooks Part III, Springer-Verlag, see p. 354 Eq. (3.11)
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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 chi(q)/chi(q^3)^3 in powers of q where chi() is a Ramanujan theta function.
Expansion of -q^(-1/3)*c(q^2)/c(-q) in powers of q where c() is a cubic AGM analog function.
Expansion of q^(-1/3)* (eta(q^2)^2* eta(q^3)^3* eta(q^12)^3)/ (eta(q)* eta(q^4)* eta(q^6)^6) in powers of q.
Given g.f. A(x), then B(x)=x*A(x^3) satisfies 0=f(B(x), B(x^2)) where f(u, v)= (1-u^3)* (1-v^3)- (1+2*u*v)* (1-u*v)^3.
Given g.f. A(x), then B(x)=x*A(x^3) satisfies 0=f(B(x), B(x^3)) where f(u, v)= (v+v^2+v^3)- u^3*(1-2*v+4*v^2).
Given g.f. A(x), then B(x)=x*A(x^3) satisfies 0=f(B(x), B(x^7)) where f(u, v)= (u-v)^8 -u*(1-u^3)* (1+8*u^3)* v*(1-v^3)* (1+8*v^3).
Given g.f. A(x), then B(x)=x*A(x^3) satisfies 0=f(B(x), B(x^2), B(x^4)) where f(u, v, w)= (v^2-u)* (v^2-w)- v^2*(v-u^2)* (v-w^2).
Given g.f. A(x), then B(x)=x*A(x^3) satisfies 0=f(B(x), B(x^2), B(x^3), B(x^6)) where f(u1, u2, u3, u6)= u6+u3-u1*u2 +u6*u3*(1-2*u1*u2).
Euler transform of period 12 sequence [ 1, -1, -2, 0, 1, 2, 1, 0, -2, -1, 1, 0, ...].
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EXAMPLE
| q + q^4 - 2*q^10 - 2*q^13 + q^16 + 4*q^19 + 4*q^22 - q^25 - 8*q^28 + ...
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PROG
| (PARI) {a(n)=local(A); if(n<0, 0, A=x*O(x^n); polcoeff( eta(x^2+A)^2*eta(x^3+A)^3*eta(x^12+A)^3/eta(x+A)/eta(x^4+A)/eta(x^6+A)^6, n))}
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CROSSREFS
| A092848(n) = (-1)^n * a(n). Convolution inverse of A062244.
Sequence in context: A110090 A196831 A092848 * A107356 A124725 A106522
Adjacent sequences: A128108 A128109 A128110 * A128112 A128113 A128114
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KEYWORD
| sign
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AUTHOR
| Michael Somos, Feb 14 2007
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