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A097197
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Expansion of q^(-1/3) eta(q^6)^2/(eta(q) eta(q^3)) in powers of q.
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3
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1, 1, 2, 4, 6, 9, 14, 20, 29, 42, 58, 80, 110, 148, 198, 264, 347, 454, 592, 764, 982, 1257, 1598, 2024, 2554, 3206, 4010, 5000, 6208, 7684, 9484, 11664, 14306, 17501, 21346, 25972, 31526, 38170, 46112, 55588, 66861, 80258, 96154, 114968, 137212, 163472, 194406
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,3
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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 psi(q^3)/f(-q) in powers of q where psi(), f() are Ramanujan theta functions.
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REFERENCES
| N. J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; p. 53, Eq. (25.95).
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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 6 sequence [ 1, 1, 2, 1, 1, 0, ...]. - Michael Somos Aug 19 2006
G.f.: (Sum_{k>=0} x^(3(k^2+k)/2))/(Product_{k>0} 1-x^k).
G.f.: (Sum_{k>0} x^(3(k^2-k)/2))/((1-x)(1-x^2)...) = Product_{k>0} (1+x^(3k))(1-x^(6k))/(1-x^k).
G.f.: Product_{k>0} (1 + x^k + x^(2*k)) * (1 + x^(3*k))^2. - Michael Somos, Apr 10 2008
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EXAMPLE
| q + q^4 + 2*q^7 + 4*q^10 + 6*q^13 + 9*q^16 + 14*q^19 + 20*q^22 + 29*q^25 + ...
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PROG
| (PARI) {a(n)=local(A); if(n<0, 0, A=x*O(x^n); polcoeff( eta(x^6+A)^2/eta(x+A)/eta(x^3+A), n))} /* Michael Somos Aug 19 2006 */
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CROSSREFS
| A139135(n) = (-1)^n * a(n).
Sequence in context: A069916 A153140 A139135 * A119737 A038718 A042942
Adjacent sequences: A097194 A097195 A097196 * A097198 A097199 A097200
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KEYWORD
| nonn
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com), Sep 17 2004; edited May 15 2008 at the suggestion of R. J. Mathar.
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