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A055978
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A sequence related to Ramanujan's tau function.
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0
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1, -2, 0, 4, -24, 36, 0, -64, 252, -290, 0, 396, -1472, 1380, 0, -944, 4830, -4248, 0, -1268, -6048, 8040, 0, 12528, -16744, -3706, 0, -20976, 84480, -31284, 0, -31312, -113643, 101542, 0, 152892, -115920, -104792, 0, -96576, 534612, -112914, 0, -369544, -370944, 334864, 0, 603936, -577738, -22554, 0
(list; graph; refs; listen; history; internal format)
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OFFSET
| 4,2
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REFERENCES
| Eichler and Zagier, The Theory of Jacobi Forms, Birkhauser, 1985.
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FORMULA
| a(4n+2)=0, a(4n)=A000594(n) (Ramanujan tau(n)).
Sum_{k>0} a(4k+1)q^(4k+1) = (-1)(q*d/dq theta_2(q^4))*eta(q^4)^18*eta(q^16)^2/eta(q^8). - Michael Somos Mar 20 2004
Sum_{k>0} a(4k+3)q^(4k+3) = (1/2)(q*d/dq theta_3(q^4))*eta(q^4)^16*eta(q^8)^5/eta(q^16)^2. - Michael Somos Mar 20 2004
G.f.: x^3(Product_{k>0} (1-x^k)(1-x^(4k))^18/(1+x^k))(Sum_{k>0} k^2 x^(k^2)). - Michael Somos Mar 20 2004
phi_{10, 1}*q*(d/dq){theta_3(z)} where phi_{10, 1} is unique Jacobi cusp form of weight 10 index 1 given by A003784.
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PROG
| (PARI) a(n)=if(n<3, 0, n-=3; X=x+x*O(x^n); polcoeff(eta(X)^2*eta(X^4)^18/eta(X^2)*sum(k=1, sqrtint(n), k^2*x^(k^2)), n))
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CROSSREFS
| A003784, A000594.
Sequence in context: A111938 A167341 A199852 * A069025 A145962 A066442
Adjacent sequences: A055975 A055976 A055977 * A055979 A055980 A055981
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KEYWORD
| sign
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AUTHOR
| Kok Seng Chua (chuaks(AT)ihpc.nus.edu.sg), Jul 24 2000
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