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Search: id:A002171
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| A002171 |
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Glaisher's chi numbers.
(Formerly M0745 N0280)
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+0
3
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| 1, -2, -3, 6, 2, 0, -1, -10, 0, -2, 10, 6, -7, 14, 0, -10, -12, 0, -6, 0, 9, -4, 10, 0, 18, -2, 0, 6, -14, -18, -11, 12, 0, 0, -22, 0, 20, 14, -6, 22, 0, 0, 23, -26, 0, -18, 4, 0, -14, -2, 0, -20, 0, 0, 0, 12, 3, 30, 26, 0, -30, 14, 0, 0, 2, 30, -28, -26, 0, -18, 10, 0, -13, -34, 0, 0, 20, 0, 26, 22, 0, -6, 0, 6, 18, 0
(list; graph; listen)
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OFFSET
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0,2
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COMMENT
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Euler transform of period 2 sequence [ -2,-4,...].
Expansion of q^-1*eta(q^4)^2*eta(q^8)^2 in powers of q^4.
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REFERENCES
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J. W. L. Glaisher, On the function chi(n), Quarterly Journal of Pure and Applied Mathematics, 20 (1884), 97-167.
Ishikawa, T., Congruences between binomial coefficients binom(2f,f) and Fourier coefficients of certain eta-products, Hiroshima Math. J. 22 (1992), no. 3, 583-590.
M. Koike, On McKay's conjecture, Nagoya Math. J., 95 (1984), 85-89.
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
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T. D. Noe, Table of n, a(n) for n=0..1000
S. R. Finch, Powers of Euler's q-Series, (arXiv:math.NT/0701251).
W. Stein, Modular Forms Database.
Index entries for sequences related to Glaisher's numbers
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FORMULA
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Also (Sum_{n>=0} (-1)^n*(2*n+1)*x^(2*n+1)^2)(Sum_n (-1)^n*x^(2n)^2).
a(n)=b(4n+1) where b(n) is multiplicative and b(p^e)=b(p)b(p^(e-1))-p*b(p^(e-2)) and b(p) = p - number of solutions of y^2=x^3-x mod p. - Michael Somos Jul 27 2006
G.f.: (Product_{k>0} (1-x^k)(1-x^(2k)))^2.
Coefficients of L-series for elliptic curve "32a2": y^2 = x^3 - x.
G.f. is Fourier series of a weight 2 level 32 modular form. f(-1/ (32 t)) = 32 (t/i)^2 f(t) where q = exp(2 pi i t).
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EXAMPLE
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eta(q^4)^2*eta(q^8)^2 = q-2*q^5-3*q^9+6*q^13+2*q^17+...
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PROGRAM
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(PARI) a(n)=ellak(ellinit([0, 0, 0, -1, 0]), 4*n+1) /* Michael Somos Jul 27 2006 */
(PARI) {a(n)=local(A); if(n<0, 0, A=x*O(x^n); polcoeff( (eta(x+A)*eta(x^2+A))^2, n))} /* Michael Somos Jul 27 2006 */
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CROSSREFS
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Cf. A002172.
Sequence in context: A124795 A084459 A093095 this_sequence A138515 A107410 A132041
Adjacent sequences: A002168 A002169 A002170 this_sequence A002172 A002173 A002174
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KEYWORD
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sign,easy,nice
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AUTHOR
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N. J. A. Sloane (njas(AT)research.att.com).
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