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A035170
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Coefficients in expansion of Dirichlet series Product_p (1-(Kronecker(m,p)+1)*p^(-s)+Kronecker(m,p)*p^(-2s))^(-1) for m = -20.
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7
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1, 1, 2, 1, 1, 2, 2, 1, 3, 1, 0, 2, 0, 2, 2, 1, 0, 3, 0, 1, 4, 0, 2, 2, 1, 0, 4, 2, 2, 2, 0, 1, 0, 0, 2, 3, 0, 0, 0, 1, 2, 4, 2, 0, 3, 2, 2, 2, 3, 1, 0, 0, 0, 4, 0, 2, 0, 2, 0, 2, 2, 0, 6, 1, 0, 0, 2, 0, 4, 2, 0, 3, 0, 0, 2, 0, 0, 0, 0, 1, 5, 2, 2, 4, 0, 2, 4, 0, 2, 3, 0, 2, 0, 2, 0, 2, 0, 3, 0, 1, 2, 0, 2, 0, 4
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,3
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COMMENTS
| Ramanujan theta functions: f(q) := Prod_{k>=1} (1-(-q)^k) (see A121373), phi(q) := theta_3(q) := Sum_{k=-oo..oo} q^(k^2) (A000122), psi(q) := Sum_{k=0..oo} q^(k*(k+1)/2) (A10054), chi(q) := Prod_{k>=0} (1+q^(2k+1)) (A000700).
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REFERENCES
| B. C. Berndt, Ramanujan's Notebooks Part III, Springer-Verlag, see p. 253.
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LINKS
| M. Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
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FORMULA
| Multiplicative with a(2^e) = a(5^e) = 1, a(p^e) = e+1 if p == 1, 3, 7, 9 (mod 20), a(p^e) = (1+(-1)^e)/2 if p == 11, 13, 17, 19 (mod 20). - Michael Somos Sep 10 2005
G.f.: Sum_{k>0} x^k(1+x^(2k))(1+x^(6k))/(1+x^(10k)) . - Michael Somos Sep 10 2005
a(2n)=a(5n)=a(n), a(20n+11)=a(20n+13)=a(20n+17)=a(20n+19)=0.
Moebius transform is period 20 sequence [ 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, -1, 0, -1, 0, 0, 0, -1, 0, -1, 0, ...]. - Michael Somos Oct 21 2006
Expansion of -1 +(phi(q)*phi(q^5) +phi(q^2)*phi(q^10) +4*q^3*psi(q^4)*psi(q^20))/2 in powers of q where phi(),psi() are Ramanujan theta functions.
2*a(n)=A028586(n)+A033718(n) if n>0. - Michael Somos Oct 21 2006
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PROG
| (PARI) direuler(p=2, 101, 1/(1-(kronecker(m, p)*(X-X^2))-X))
(PARI) a(n)=if(n<1, 0, sumdiv(n, d, kronecker(-20, d))) /* Michael Somos Sep 10 2005 */
(PARI) a(n)=if(n<1, 0, direuler(p=2, n, 1/(1-X)/ (1-kronecker(-20, p)*X) )[n]) /* Michael Somos Sep 10 2005 */
(PARI) {a(n)=if(n<1, 0, qfrep([1, 0; 0, 5], n)[n] +qfrep([2, 1; 1, 3], n)[n])} /* Michael Somos Oct 21 2006 */
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CROSSREFS
| |A111949(n)|=a(n).
Sequence in context: A124233 * A111949 A143323 A086598 A074746 A133188
Adjacent sequences: A035167 A035168 A035169 * A035171 A035172 A035173
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KEYWORD
| nonn,mult
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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