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Search: id:A035016
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| A035016 |
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Fourier coefficients of E_{0,4}. |
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+0
4
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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; listen)
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OFFSET
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0,2
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COMMENT
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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
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N. J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; p. 77, Eq. (31.61).
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LINKS
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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
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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
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EXAMPLE
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1-16q+112q^2- ...
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PROGRAM
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(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))}
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CROSSREFS
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Cf. A000143 (same but without signs).
Sequence in context: A053526 A107908 A000143 this_sequence A081194 A121148 A091031
Adjacent sequences: A035013 A035014 A035015 this_sequence A035017 A035018 A035019
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KEYWORD
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sign,easy,nice
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AUTHOR
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Barry Brent (barryb(AT)primenet.com)
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