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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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2
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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; refs; listen; history; internal format)
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OFFSET
| 1,8
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COMMENTS
| Unique cusp form of weight 2 for congruence group Gamma_1(15). - Michael Somos, Aug 11 2011
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
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REFERENCES
| 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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LINKS
| M. D. Rogers, Hypergeometric formulas for lattice sums and Mahler measures.
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FORMULA
| 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
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
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
G.f.: x Product_{k>0} (1 - x^k) * (1 - x^(3*k)) * (1 - x^(5*k)) * (1 - x^(15*k)).
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EXAMPLE
| q - q^2 - q^3 - q^4 + q^5 + q^6 + 3*q^8 + q^9 - q^10 - 4*q^11 + q^12 - 2*q^13 - ...
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MATHEMATICA
| a[ n_] := SeriesCoefficient[ q QPochhammer[ q, q] QPochhammer[ q^3, q^3] QPochhammer[ q^5, q^5] QPochhammer[ q^15, q^15], {q, 0, n}] (* Michael Somos, Aug 11 2011 *)
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PROG
| (PARI) {a(n) = if( n<1, 0, ellak( ellinit([ 1, 1, 1, 0, 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==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 */
(SAGE) CuspForms( Gamma1(15), 2, prec = 100). 0 # Michael Somos, Aug 11 2011
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CROSSREFS
| Sequence in context: A111956 A024564 A084795 * A104610 A138684 A132442
Adjacent sequences: A030181 A030182 A030183 * A030185 A030186 A030187
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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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