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A030210
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Expansion of (eta(q) * eta(q^5))^4 in powers of q.
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1
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1, -4, 2, 8, -5, -8, 6, 0, -23, 20, 32, 16, -38, -24, -10, -64, 26, 92, 100, -40, 12, -128, -78, 0, 25, 152, -100, 48, -50, 40, -108, 256, 64, -104, -30, -184, 266, -400, -76, 0, 22, -48, 442, 256, 115, 312, -514, -128, -307, -100, 52, -304, 2, 400, -160, 0, 200, 200, 500, -80, -518, 432, -138, -512
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,2
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COMMENTS
| Conjecture: |a(p)| < 2*p^(3/2) for p prime. - Michael Somos, Oct 31 2005
Unique cusp form of weight 4 for congruence group Gamma_1(5). - Michael Somos, Aug 11 2011
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REFERENCES
| M. Koike, On McKay's conjecture, Nagoya Math. J., 95 (1984), 85-89.
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FORMULA
| Euler transform of period 5 sequence [ -4, -4, -4, -4, -8, ...]. - Michael Somos, May 02 2005
G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^4)) where f(u, v, w) = -v^3 + 8*u*v*w + 16*u*w^2 + u^2*w. - Michael Somos, May 02 2005
a(n) is multiplicative and a(5^e) = (-5)^e, a(p^(e+2)) = a(p) * a(p^(e+1)) - p^3 * a(p^e).
G.f. is a period 1 Fourier series which satisfies f(-1 / (5 t)) = 25 (t/i)^4 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^(5*k)))^4.
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EXAMPLE
| q - 4*q^2 + 2*q^3 + 8*q^4 - 5*q^5 - 8*q^6 + 6*q^7 - 23*q^9 + 20*q^10 + 32*q^11 + ...
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MATHEMATICA
| a[ n_] := SeriesCoefficient[ q (QPochhammer[ q, q] QPochhammer[ q^5, q^5])^4, {q, 0, n}] (* Michael Somos, Aug 11 2011 *)
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PROG
| (PARI) {a(n) = local(A); if( n<1, 0, n--; A = x^n * O(x); polcoeff( (eta(x + A) * eta(x^5 + A))^4, n))}
(SAGE) CuspForms( Gamma1(5), 4, prec = 100). 0 # Michael Somos, Aug 11 2011
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CROSSREFS
| Sequence in context: A030181 A021879 A020806 * A098798 A131783 A094312
Adjacent sequences: A030207 A030208 A030209 * A030211 A030212 A030213
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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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