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Search: id:A136747
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| A136747 |
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Expansion of newform of level 3 weight 8 and trivial character. |
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+0
1
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| 1, 6, -27, -92, 390, -162, -64, -1320, 729, 2340, -948, 2484, -5098, -384, -10530, 3856, 28386, 4374, -8620, -35880, 1728, -5688, -15288, 35640, 73975, -30588, -19683, 5888, 36510, -63180, -276808, 192096, 25596, 170316, -24960, -67068, 268526, -51720, 137646, -514800
(list; graph; listen)
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OFFSET
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1,2
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LINKS
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W. Stein, Modular Forms Database.
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FORMULA
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Expansion of a(q)^2 * (b(q) * c(q) / 3)^3 in powers of q where a(), b(), c() are cubic AGM functions.
Expansion of (eta(q) * eta(q^3))^6 * ((eta(q)^3 + 9 * eta(q^9)^3) / eta(q^3) )^2 in powers of q.
a(n) is multiplicative with a(3^e) = (-27)^e, a(p^e) = a(p) * a(p^(e-1)) - p^7 * a(p^(e-2)) unless p = 3.
G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^4)) where f(u, v, w) = u^4*w +512*u^3*w^2 +131072*u^2*w^3 +16777216*u*w^4 -24*u^3*v*w -9216*u^2*v*w^2 -1572864*u*v*w^3 +288*u^2*v^2*w +73728*u*v^2*w^2 -u^2*v^3 -1984*w*v^3*u -65536*w^2*v^3 +12*v^4*u +3072*w*v^4 -36*v^5.
G.f. is a period 1 Fourier series that satisfies f(-1 / (3 t)) = 81 (t/i)^8 f(t) where q = exp(2 pi i t).
G.f.: x * (prod_{k>0} (1 - x^k) * (1 - x^(3*k)))^6 * (sum_{j,k} x^(j*j + j*k + k*k))^2.
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EXAMPLE
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q + 6*q^2 - 27*q^3 - 92*q^4 + 390*q^5 - 162*q^6 - 64*q^7 - 1320*q^8 + ...
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PROGRAM
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(PARI) {a(n) = local(A); if( n<1, 0, n--; A = x^n * O(x); polcoeff( (eta(x + A) * eta(x^3 + A))^6 * sum(k=1, n, 12 * (sigma(3*k) - 3 * sigma(k)) * x^k, 1 + A), n))}
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CROSSREFS
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Convolution of A007332 and A008653.
Sequence in context: A052267 A038166 A121596 this_sequence A001940 A121591 A071734
Adjacent sequences: A136744 A136745 A136746 this_sequence A136748 A136749 A136750
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KEYWORD
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sign
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AUTHOR
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Michael Somos, Jan 20 2008
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