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A030187
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Expansion of eta(q) * eta(q^2) * eta(q^7) * eta(q^14) in powers of q.
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2
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1, -1, -2, 1, 0, 2, 1, -1, 1, 0, 0, -2, -4, -1, 0, 1, 6, -1, 2, 0, -2, 0, 0, 2, -5, 4, 4, 1, -6, 0, -4, -1, 0, -6, 0, 1, 2, -2, 8, 0, 6, 2, 8, 0, 0, 0, -12, -2, 1, 5, -12, -4, 6, -4, 0, -1, -4, 6, -6, 0, 8, 4, 1, 1, 0, 0, -4, 6, 0, 0, 0, -1, 2, -2, 10, 2, 0, -8, 8, 0, -11, -6, -6, -2, 0, -8, 12, 0, -6, 0, -4, 0, 8, 12, 0, 2, -10, -1, 0, -5, 0
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,3
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COMMENTS
| Associated with permutations in Mathieu group M24 of shape (14)(7)(2)(1).
Coefficients of L-series for elliptic curve "14a4": y^2 + x*y + y = x^3 - x or y^2 + x*y - y = x^3. - Michael Somos, Feb 19 2007
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REFERENCES
| M. Koike, Matheiu group M24 and modular forms, Nagoya Math. J., 99 (1985), 147-157. MR0805086 (87e:11060)
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FORMULA
| Euler transform of period 14 sequence [ -1, -2, -1, -2, -1, -2, -2, -2, -1, -2, -1, -2, -1, -4, ...]. - Michael Somos, Aug 13 2006
a(n) is multiplicative with a(2^e) = (-1)^e, a(7^e) = 1, otherwise a(p^e) = a(p) * a(p^(e-1)) - p * a(p^(e-2)) where a(p) = p minus number of points of elliptic curve modulo p . - Michael Somos, Aug 13 2006
G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^4)) where f(u, v, w )= v^4 - u*w* (u + 2*v) * (v + 2*w). - Michael Somos, Feb 19 2007
G.f. is a period 1 Fourier series which satisfies f(-1 / (14 t)) = 14 (t/i)^2 g(t) where q = exp(2 pi i t). - Michael Somos, Aug 11 2011
G.f.: x Product_{k>0} (1 - x^k) * (1 - x^(2*k)) * (1 - x^(7*k)) * (1 - x^(14*k)).
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EXAMPLE
| q - q^2 - 2*q^3 + q^4 + 2*q^6 + q^7 - q^8 + q^9 - 2*q^12 - 4*q^13 - q^14 + q^16 + ...
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MATHEMATICA
| a[ n_] := SeriesCoefficient[ q QPochhammer[ q, q] QPochhammer[ q^2, q^2] QPochhammer[ q^7, q^7] QPochhammer[ q^14, q^14], {q, 0, n}] (* Michael Somos, Aug 11 2011 *)
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PROG
| (PARI) {a(n) = if( n<1, 0, ellak( ellinit([ -1, 0, -1, -1, 0], 1), n))} /* Michael Somos, Aug 13 2006 */
(PARI) {a(n) = local(A, p, e, x, y, a0, a1); if( n<1, 0, A = factor(n); prod( k=1, matsize(A)[1], if( p=A[k, 1], e=A[k, 2]; if( p==2, (-1)^e, if( p==7, 1, a0=1; a1 = y = -sum( x=0, p-1, kronecker( 4*x^3+x^2-2*x+1, p)); for( i=2, e, x = y*a1 - p*a0; a0=a1; a1=x); a1)))))} /* Michael Somos, Aug 13 2006 */
(PARI) {a(n) = local(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( eta(x + A) * eta(x^2 + A) * eta(x^7 + A) * eta(x^14 + A), n))}
(SAGE) CuspForms( Gamma1(14), 2, prec = 100). 0 # Michael Somos, Aug 11 2011
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CROSSREFS
| Sequence in context: A024361 A190676 A135486 * A117278 A140082 A025852
Adjacent sequences: A030184 A030185 A030186 * A030188 A030189 A030190
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KEYWORD
| sign,mult
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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