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A001934
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Expansion of 1/theta_4(q)^2 in powers of q.
(Formerly M3443 N1397)
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3
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1, 4, 12, 32, 76, 168, 352, 704, 1356, 2532, 4600, 8160, 14176, 24168, 40512, 66880, 108876, 174984, 277932, 436640, 679032, 1046016, 1597088, 2418240, 3632992, 5417708, 8022840, 11802176, 17252928, 25070568, 36223424, 52053760, 74414412
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,2
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COMMENTS
| Euler transform of period 2 sequence [ 4, 2, ...].
The Cayley reference actually is to A004403. - Michael Somos Feb 24 2011
Number of overpartition pairs, see Lovejoy reference. [Joerg Arndt, Apr 3 2011]
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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.
B. Kim, Overpartition pairs modulo powers of 2, Discrete Math., 311 (2011), 835-840.
Jeremy Lovejoy, Overpartition pairs, Annales de l'institut Fourier, vol.56, no.3, p.781-794, 2006.
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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FORMULA
| G.f.: Product ( 1 - x^k )^{-c(k)}, c(k) = 4, 2, 4, 2, 4, 2, ....
G.f. prod{i>=1, (1+x^i)^2/(1-x^i)^2} - Jon Perry (perry(AT)globalnet.co.uk), Apr 04 2004
Expansion of eta(q^2)^2/eta(q)^4 in powers of q, where eta(x)=prod(n>=1,1-q^n).
a(n) = (-1)^n * A004403(n). a(n) = 4 * A002318(n) unless n=0. - Michael Somos Feb 24 2011
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MAPLE
| mul((1+x^n)^2/(1-x^n)^2, n=1..256);
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MATHEMATICA
| CoefficientList[Series[1/EllipticTheta[4, 0, q]^2, {q, 0, 32}], q] (* From Jean-François Alcover, Jul 18 2011 *)
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PROG
| (PARI) y=prod(i=1, 20, (1+x^i)^2)/prod(i=1, 20, (1-x^i)^2); for(i=0, 20, print1(", "polcoeff(y, i)))
(PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^2 / eta(x + A)^4, n))} /* Michael Somos Feb 09 2006 */
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CROSSREFS
| Cf. A004403, A002318.
Sequence in context: A127811 A138517 * A004403 A084566 A079769 A107035
Adjacent sequences: A001931 A001932 A001933 * A001935 A001936 A001937
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KEYWORD
| nonn,easy
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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
| More terms from James A. Sellers (sellersj(AT)math.psu.edu), Sep 08 2000
Edited by N. J. A. Sloane (njas(AT)research.att.com) May 13 2008 to remove an incorrect g.f.
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