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A092848
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Expansion of reciprocal of Hauptmodul for Gamma_0(18).
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15
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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. 345 Entry 1(i).
W. Duke, Continued fractions and modular functions, Bull. Amer. Math. Soc. 42 (2005), 137-162. See page 155 Eq. (9.13)
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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^(2k-1))/(1-x^(6k-3))^3.
Given g.f. A(x), then B(x)=x*A(x^3) satisfies 0=f(B(x), B(x^2)) where f(u, v)=u^2-v+2uv^2.
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^3-v^2+v)-u^3(1+2v+4v^2).
Expansion of q^(-1/3)eta(q)eta(q^6)^3/(eta(q^2)eta(q^3)^3) in powers of q.
Euler transform of period 6 sequence [ -1,0,2,0,-1,0,...].
Expansion of chi(-q)/chi(-q^3)^3 where chi() (g.f. A000700) is a Ramanujan theta function.
G.f.: 1/(1+ (x+x^2)/(1+ (x^2+x^4)/(1 +(x^3+x^6)/...))).
Expansion of q^(-1/3)c(q^2)/c(q) where c() is a cubic AGM analog function. - Michael Somos Oct 04 2006
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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, m); if(n<0, 0, A=1+O(x); m=1; while(m<=n, m*=2; A=subst(A, x, x^2); A=sqrt(A+(x*A^2)^2)-x*A^2); polcoeff(A, n))}
(PARI) {a(n)=if(n<0, 0, polcoeff(prod(k=0, (n-1)\2, (1-x^(2*k+1))^if(k%3==1, -2, 1), 1+x*O(x^n)), n))}
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CROSSREFS
| A062242(2*n + 1) = a(n). A128111(n) = (-1)^n * a(n). Convolution inverse of A062242.
Sequence in context: A110090 A196831 * A128111 A107356 A124725 A106522
Adjacent sequences: A092845 A092846 A092847 * A092849 A092850 A092851
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KEYWORD
| sign
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AUTHOR
| Michael Somos, Mar 07 2004
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