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Search: id:A030200
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| A030200 |
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Expansion of q^(-1/2) * eta(q) * eta(q^11) in powers of q. |
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2
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| 1, -1, -1, 0, 0, 1, 0, 1, 0, 0, 0, -1, 0, 1, 0, -1, -1, 0, -1, 0, 0, 0, 0, 2, 1, 0, 2, -1, 0, -1, 0, 0, 0, -1, 1, -1, 0, 0, 0, 0, -1, 0, 0, 0, -1, 0, 1, 0, -1, 0, 0, 2, 0, 0, 0, 1, -1, 1, 0, 0, 1, 0, 1, 0, 0, 0, 0, -1, -1, 0, -2, 0, 0, -1, 0, 0, 0, 1, -1, -2, 0, 2, 1, 0, 1, 0, 0, 0, 1, -1, -1, 0, 1, 0, 0, -1, 0, 0, 0, 2, 1, 0, 0, 0, 0
(list; graph; listen)
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OFFSET
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0,24
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REFERENCES
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M. Koike, On McKay's conjecture, Nagoya Math. J., 95 (1984), 85-89.
H. McKean and V. Moll, Elliptic Curves, Cambridge University Press, 1997, page 203. MR1471703 (98g:14032)
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FORMULA
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Euler transform of period 11 sequence [ -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -2, ...]. - Michael Somos Nov 20 2006
a(n)=b(2n+1) where b(n) is multiplicative with b(2^e) = 0^e, b(11^e) = 1, b(p^e) = (e-1)%3-1 if f=0, b(p^e) = e+1 if f=3, b(p^e) = (1+(-1)^e)/2 if f=1 where f = number of zeros of x^3-x^2-x-1 modulo p. - Michael Somos Nov 20 2006
G.f.: Product_{k>0} (1-x^k)*(1-x^(11k)).
Sum over all solutions to x^2+xy+3y^2=2n+1 with x>0 odd of (-1)^y. - Michael Somos Jan 29 2007
G.f. is a period 1 Fourier series which satisfies f(-1 / (11 t)) = 11^(1/2) (t/i) f(t) where q = exp(2 pi i t).
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EXAMPLE
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q - q^3 - q^5 + q^11 + q^15 - q^23 + q^27 - q^31 - q^33 - q^37 + 2*q^47 +...
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PROGRAM
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(PARI) {a(n)= if(n<0, 0, n=2*n+1; qfrep([1, 0; 0, 11], n)[n] -qfrep([3, 1; 1, 4], n)[n])} /* Michael Somos Nov 20 2006 */
(PARI) {a(n)= local(A, p, e, f); if(n<0, 0, n=2*n+1; A=factor(n); prod(k=1, matsize(A)[1], if(p=A[k, 1], e=A[k, 2]; if(p==2, 0, if(p==11, 1, f = sum(k=0, p-1, (k^3-k^2-k-1)%p==0); if(f==0, (e-1)%3-1, if(f==1, (1+(-1)^e)/2, e+1)))))))} /* Michael Somos Nov 20 2006 */
(PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A) * eta(x^11 + A), n))} /* Michael Somos Nov 20 2006 */
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CROSSREFS
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Cf. A106276. Convolution square is A006571.
Sequence in context: A030204 A138514 A143540 this_sequence A095734 A137269 A112201
Adjacent sequences: A030197 A030198 A030199 this_sequence A030201 A030202 A030203
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KEYWORD
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sign
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AUTHOR
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N. J. A. Sloane (njas(AT)research.att.com).
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