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Search: id:A001938
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| A001938 |
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Expansion of k/(4q^(1/2)) in powers of q, where k is the elliptic function defined by sqrt(k) = theta_2/theta_3.
(Formerly M3475 N1412)
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+0
10
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| 1, -4, 14, -40, 101, -236, 518, -1080, 2162, -4180, 7840, -14328, 25591, -44776, 76918, -129952, 216240, -354864, 574958, -920600, 1457946, -2285452, 3548550, -5460592, 8332425, -12614088, 18953310, -28276968, 41904208, -61702876, 90304598, -131399624
(list; graph; listen)
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OFFSET
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0,2
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REFERENCES
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N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
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.
E. T. Copson, An Introduction to the Theory of Functions of a Complex Variable, 1935, Oxford University Press, p. 385.
N. J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; Eq. (34.3).
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LINKS
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T. D. Noe, Table of n, a(n) for n=0..1000
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FORMULA
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G.f.: (Product_{n>0} (1+x^(2n))/(1+x^(2n-1)))^4.
G.f. A(x) satisfies 1=(1-16xA(x)^2)(1+16xA(-x)^2). - Michael Somos, Mar 26 2004
Given g.f. A(x), then B(x)=A(x^2)x satisfies 0=f(B(x), B(x^2)) where f(u, v)=v-(u(1+4v))^2. - Michael Somos, Mar 26 2004
Expansion of q^(-1/2)* (eta(q)* eta(q^4)^2/ eta(q^2)^3)^4 in powers of q.
Euler transform of period 4 sequence [ -4, 8, -4, 0, ...].
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PROGRAM
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(PARI) {a(n)=local(A, A2, m); if(n<0, 0, n=2*n+1; A=x+O(x^3); m=2; while(m<n, m*=2; A=subst(A, x, x^2); A=sqrt(A)/(1+4*A)); polcoeff(A, n))} - Michael Somos, Mar 26 2004
(PARI) {a(n)= local(A); if(n<0, 0, A= x*O(x^n); polcoeff( (eta(x+A)* eta(x^4+A)^2/ eta(x^2+A)^3)^4, n))} - Michael Somos, Mar 26 2004
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CROSSREFS
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Cf. A127931, A127932, A127393, A079006, A001936.
a(n) = (-1)^n* A093160(n).
Sequence in context: A144141 A066375 A093160 this_sequence A066368 A121593 A023003
Adjacent sequences: A001935 A001936 A001937 this_sequence A001939 A001940 A001941
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KEYWORD
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sign,nice
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AUTHOR
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N. J. A. Sloane (njas(AT)research.att.com).
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EXTENSIONS
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Edited by N. J. A. Sloane (njas(AT)research.att.com), Mar 31 2007
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