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A001937
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Expansion of (psi(x^2) / psi(-x))^3 in powers of x where psi() is a Ramanujan theta function.
(Formerly M2785 N1120)
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2
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1, 3, 9, 22, 48, 99, 194, 363, 657, 1155, 1977, 3312, 5443, 8787, 13968, 21894, 33873, 51795, 78345, 117312, 174033, 255945, 373353, 540486, 776848, 1109040, 1573209, 2218198, 3109713, 4335840, 6014123, 8300811, 11402928, 15593702, 21232521
(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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REFERENCES
| A. Cayley, A memoir on the transformation of elliptic functions, Collected Mathematical Papers. Vols. 1-13, Cambridge Univ. Press, London, 1889-1897, Vol. 9, p. 128.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
| Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
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FORMULA
| Euler transform of period 4 sequence [ 3, 3, 3, 0, ...]. - Michael Somos Mar 06 2011
Convolution cube of A001935. A187053(n) = (-1)^n * a(n). - Michael Somos Mar 06 2011
G.f.: (Product_{k>0} (1 + x^(2*k)) / (1 - x^(2*k-1)))^3.
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MATHEMATICA
| CoefficientList[ Series[Product[(1 - x^k)^(-3*Boole[Mod[k, 4] != 0]), {k, 1, 101}], {x, 0, 100}], x] [From Olivier GERARD (olivier.gerard(AT)gmail.com), May 06 2009]
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PROG
| (PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x^4 + A) / eta(x + A))^3, n))} /* Michael Somos Mar 06 2011 */
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CROSSREFS
| Cf. A001936, A079006, A001935, A083365, A001938, A093160, A001939, A001940, A001941. [From Olivier GERARD (olivier.gerard(AT)gmail.com), May 06 2009]
Sequence in context: A064808 A192389 A187053 * A086817 A000715 A034505
Adjacent sequences: A001934 A001935 A001936 * A001938 A001939 A001940
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KEYWORD
| nonn
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com), Simon Plouffe (simon.plouffe(AT)gmail.com)
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EXTENSIONS
| Corrected and extended by Simon Plouffe (simon.plouffe(AT)gmail.com)
Checked and more terms from Olivier GERARD (olivier.gerard(AT)gmail.com), May 06 2009
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