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A035016
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Fourier coefficients of E_{0,4}.
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5
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1, -16, 112, -448, 1136, -2016, 3136, -5504, 9328, -12112, 14112, -21312, 31808, -35168, 38528, -56448, 74864, -78624, 84784, -109760, 143136, -154112, 149184, -194688, 261184, -252016, 246176, -327040, 390784, -390240, 395136, -476672, 599152, -596736
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,2
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COMMENTS
| E_{0,4} is unique normalized entire modular form of weight 4 for \Gamma_0(2) with a zero at zero. Also |a(n)| matches expansion of theta_3(z)^8 (A000143).
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REFERENCES
| N. J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; p. 77, Eq. (31.61).
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LINKS
| T. D. Noe, Table of n, a(n) for n=0..1000
Borcherds, Richard E., Automorphic forms with singularities on Grassmannians, Invent. Math. 132 (1998), 491-562.
B. Brent, Quadratic Minima and Modular Forms, Experimental Mathematics, v.7 no.3, 257-274.
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FORMULA
| a(0)=1; for n>0, a(n) = 16*sum_{0<d|n}(-1)^d d^3.
G.f.: Product_{n>=1} ((1-q^n)/(1+q^n))^8 [Fine]
Expansion of eta(q)^16/eta(q^2)^8 in powers of q.
Euler transform of period 2 sequence [ -16, -8, ...]. - Michael Somos, Apr 10 2005
G.f. A(x) satisfies 0=f(A(x), A(x^2), A(x^4)) where f(u, v, w)=v^3+uv(u-2v+16w)-16uw^2. - Michael Somos Apr 10 2005
G.f. is a period 1 Fourier series which satisfies f(-1 / (2 t)) = 256 (t / i)^4 g(t) where g() is g.f. for A007331. - Michael Somos Jan 11 2009
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EXAMPLE
| 1 - 16*q + 112*q^2 - 448*q^3 + 1136*q^4 - 2016*q^5 + 3136*q^6 - 5504*q^7 + ...
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MATHEMATICA
| a[0] = 1; a[n_] := 16*Sum[(-1)^d*d^3, {d, Divisors[n]}]; Table[a[n], {n, 0, 33}] (* From Jean-François Alcover, Feb 06 2012, after Pari *)
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PROG
| (PARI) a(n)=if(n<1, n==0, 16*sumdiv(n, d, (-1)^d*d^3))
(PARI) {a(n)=if(n<0, 0, polcoeff( prod(k=1, n, (1-x^k)/(1+x^k), 1+x*O(x^n))^8, n))}
(PARI) {a(n) = local(A); if( n<0, 0, A = x^n * O(x); polcoeff( (eta(x + A)^2 / eta(x^2 + A))^8, n))} /* Michael Somos Jan 11 2009 */
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CROSSREFS
| (-1)^n * A000143(n) = a(n).
Sequence in context: A177046 A144449 A000143 * A205964 A081194 A121148
Adjacent sequences: A035013 A035014 A035015 * A035017 A035018 A035019
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KEYWORD
| sign,easy,nice,changed
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AUTHOR
| Barry Brent (barryb(AT)primenet.com)
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