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A122865
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Expansion of chi(q)* phi(q^3)* psi(-q^3) in powers of q where chi(), phi(), psi() are Ramanujan theta functions.
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9
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1, 1, 0, 2, 2, 1, 0, 0, 3, 0, 0, 2, 2, 2, 0, 0, 1, 2, 0, 2, 2, 1, 0, 0, 2, 0, 0, 2, 4, 0, 0, 0, 2, 3, 0, 2, 2, 0, 0, 0, 1, 0, 0, 4, 0, 2, 0, 0, 4, 2, 0, 0, 2, 2, 0, 0, 3, 0, 0, 2, 2, 0, 0, 0, 2, 1, 0, 2, 4, 2, 0, 0, 0, 0, 0, 2, 2, 2, 0, 0, 2, 2, 0, 4, 0, 1, 0, 0, 4, 0, 0, 2, 2, 0, 0, 0, 3, 2, 0, 2, 0, 0, 0, 0, 2
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,4
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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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LINKS
| M. Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
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FORMULA
| Expansion of q^(-1/3)*eta(q^2)^2*eta(q^6)^4/(eta(q)*eta(q^3)*eta(q^4)*eta(q^12)) in powers of q.
Euler transform of period 12 sequence [ 1, -1, 2, 0, 1, -4, 1, 0, 2, -1, 1, -2, ...]. - Michael Somos Apr 19 2007
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PROG
| (PARI) {a(n)=local(A); if(n<0, 0, A=x*O(x^n); polcoeff( eta(x^2+A)^2*eta(x^6+A)^4/ (eta(x+A)*eta(x^3+A)*eta(x^4+A)*eta(x^12+A)), n))}
(PARI) {a(n)=local(A, p, e); if(n<0, 0, n=3*n+1; A=factor(n); prod(k=1, matsize(A)[1], if(p=A[k, 1], e=A[k, 2]; if(p==2, 1, if(p==3, -2*(-1)^e, if(p%4==1, e+1, 1-e%2))))))}
(PARI) {a(n)=if(n<0, 0, n=3*n+1; sumdiv(n, d, (d%4==1) -(d%4==3)))} /* Michael Somos Apr 19 2007 */
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CROSSREFS
| A002654(3n+1)=A035154(3n+1)=A113446(3n+1)=A122864(3n+1)=a(n).
Sequence in context: A029390 A108040 A137566 * A074080 A179769 A111407
Adjacent sequences: A122862 A122863 A122864 * A122866 A122867 A122868
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KEYWORD
| nonn
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AUTHOR
| Michael Somos, Sep 15 2006
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