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Search: id:A030184
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| A030184 |
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Expansion of eta(q)* eta(q^3)* eta(q^5)* eta(q^15) in powers of q. |
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3
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| 1, -1, -1, -1, 1, 1, 0, 3, 1, -1, -4, 1, -2, 0, -1, -1, 2, -1, 4, -1, 0, 4, 0, -3, 1, 2, -1, 0, -2, 1, 0, -5, 4, -2, 0, -1, -10, -4, 2, 3, 10, 0, 4, 4, 1, 0, 8, 1, -7, -1, -2, 2, -10, 1, -4, 0, -4, 2, -4, 1, -2, 0, 0, 7, -2, -4, 12, -2, 0, 0, -8, 3, 10, 10, -1, -4, 0, -2, 0, -1, 1, -10, 12, 0, 2, -4, 2, -12, -6, -1, 0, 0, 0, -8, 4, 5, 2, 7, -4, -1
(list; graph; listen)
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OFFSET
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1,8
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REFERENCES
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N. D. Elkies, Elliptic and modular curves..., in AMS/IP Studies in Advanced Math., 7 (1998), 21-76, esp. p. 70.
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FORMULA
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Euler transform of period 15 sequence [ -1, -1, -2, -1, -2, -2, -1, -1, -2, -2, -1, -2, -1, -1, -4, ...]. - Michael Somos, May 02 2005
G.f. x Product_{k>0} (1-x^k)(1-x^(3k))(1-x^(5k))(1-x^(15k)).
a(n) is multiplicative with a(5^e) = 1, a(3^e) = (-1)^e, 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
Coefficients of L-series for elliptic curve "15a8": y^2 +x*y +y= x^3 +x^2 or y^2 +x*y -y= x^3 +x^2 +x . - Michael Somos Feb 01 2004
G.f. A(x) satisfies 0= f(A(x), A(x^2), A(x^4)) where f(u, v, w)= v^3 -u*w* (u +2*v +4*w) . - Michael Somos, May 02 2005
G.f. A(x) satisfies 2*A(x^2)= -(A(x) +A(-x) +4*A(x^4)), A(x^2)^3 = -A(x)* A(-x)* A(x^4) . - Michael Somos Feb 19 2007
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PROGRAM
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(PARI) {a(n)=if(n<1, 0, ellak(ellinit([ 1, 1, 1, 0, 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==3, (-1)^e, if(p==5, 1, a0=1; a1=y=-if(p==2, 1, sum(x=0, p-1, kronecker(4*x^3+5*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^3+A)* eta(x^5+A)* eta(x^15+A), n))} /* Michael Somos May 02 2005 */
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CROSSREFS
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Sequence in context: A111956 A024564 A084795 this_sequence A104610 A138684 A132442
Adjacent sequences: A030181 A030182 A030183 this_sequence A030185 A030186 A030187
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KEYWORD
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sign,mult
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AUTHOR
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njas
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