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A122855
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Expansion of (phi(q^3)phi(q^5)+phi(q)phi(q^15))/2 in powers of q where phi(q) is a Ramanujan theta function.
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1, 1, 0, 1, 1, 1, 0, 0, 2, 1, 0, 0, 1, 0, 0, 1, 3, 2, 0, 2, 1, 0, 0, 2, 2, 1, 0, 1, 0, 0, 0, 2, 4, 0, 0, 0, 1, 0, 0, 0, 2, 0, 0, 0, 0, 1, 0, 2, 3, 1, 0, 2, 0, 2, 0, 0, 0, 2, 0, 0, 1, 2, 0, 0, 5, 0, 0, 0, 2, 2, 0, 0, 2, 0, 0, 1, 2, 0, 0, 2, 3, 1, 0, 2, 0, 2, 0, 0, 0, 0, 0, 0, 2, 2, 0, 2, 4, 0, 0, 0, 1, 0, 0, 0, 0
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,9
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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
A. Berkovich and H. Yesilyurt, Ramanujan's identities and representation of integers by certain binary and quaternary quadratic forms
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FORMULA
| Expansion of (eta(q^2)^2*eta(q^6)eta(q^10)eta(q^30)^2)/(eta(q)eta(q^4)eta(q^15)eta(q^60)) in powers of q.
a(n) is multiplicative with a(2^e) = |e-1|, a(3^e)=a(5^e)=1, a(p^e) = e+1 if p == 1, 2, 4, 8 (mod 15), a(p^e) = (1+(-1)^e)/2 if p == 7, 11, 13, 14 (mod 15).
Euler transform of period 60 sequence [ 1, -1, 1, 0, 1, -2, 1, 0, 1, -2, 1, -1, 1, -1, 2, 0, 1, -2, 1, -1, 1, -1, 1, -1, 1, -1, 1, 0, 1, -4, 1, 0, 1, -1, 1, -1, 1, -1, 1, -1, 1, -2, 1, 0, 2, -1, 1, -1, 1, -2, 1, 0, 1, -2, 1, 0, 1, -1, 1, -2, ...].
Moebius transform is period 60 sequence [ 1, -1, 0, 1, 0, 0, -1, 1, 0, 0, -1, 0, -1, 1, 0, 1, 1, 0, 1, 0, 0, 1, 1, 0, 0, 1, 0, -1, -1, 0, 1, 1, 0, -1, 0, 0, -1, -1, 0, 0, -1, 0, -1, -1, 0, -1, 1, 0, 1, 0, 0, -1, 1, 0, 0, -1, 0, 1, -1, 0, ...].
a(15n+7)=a(15n+11)=a(15n+13)=a(15n+14)=0. a(3n)=a(5n)=a(n).
G.f.: 1+Sum_{k>0} kronecker(-15,k) x^k/(1-(-x)^k).
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PROG
| (PARI) {a(n)=if(n<1, n==0, sumdiv(n, d, kronecker(-15, d)*(-1)^(d%4==2)))}
(PARI) {a(n)= local(A, p, e); if(n<1, n==0, A=factor(n); prod(k=1, matsize(A)[1], if(p=A[k, 1], e=A[k, 2]; if(p==2, e-1, if(p<7, 1, if(p%15==2^valuation(p%15, 2), e+1, 1-e%2))))))}
(PARI) {a(n)=local(A); if(n<0, 0, A=x*O(x^n); polcoeff( eta(x^2+A)^2*eta(x^6+A)*eta(x^10+A)*eta(x^30+A)^2/ (eta(x+A)*eta(x^4+A)*eta(x^15+A)*eta(x^60+A)), n))}
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CROSSREFS
| A035175(n)=a(4n).
Sequence in context: A086017 A000161 A060398 * A140727 A140728 A130068
Adjacent sequences: A122852 A122853 A122854 * A122856 A122857 A122858
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KEYWORD
| nonn,mult
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AUTHOR
| Michael Somos, Sep 14 2006
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