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Search: id:A030187
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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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3
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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; listen)
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OFFSET
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1,3
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REFERENCES
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M. Koike, Matheiu group M24 and modular forms, Nagoya Math. J., 99 (1985), 147-157. MR0805086 (87e:11060)
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FORMULA
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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.: x Product_{k>0} (1-x^k)(1-x^(2k))(1-x^(7k))(1-x^(14k)).
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
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
Associated with permutations in Mathieu group M24 of shape (14)(7)(2)(1).
G.f. is Fourier series of a weight 2 level 14 modular form. f(-1/ (14 t)) = 14 (t/i)^2 f(t) where q = exp(2 pi i t).
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PROGRAM
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(PARI) {a(n)=if(n<1, 0, ellak(ellinit([ -1, 0, -1, -1, 0]), 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))}
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CROSSREFS
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Sequence in context: A130182 A024361 A135486 this_sequence A117278 A140082 A025852
Adjacent sequences: A030184 A030185 A030186 this_sequence A030188 A030189 A030190
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KEYWORD
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sign,mult
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AUTHOR
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N. J. A. Sloane (njas(AT)research.att.com).
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