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A209614
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G.f.: Sum_{n>=1} Fibonacci(n^3)*x^(n^3).
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1
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1, 0, 0, 0, 0, 0, 0, 21, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 196418, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 10610209857723, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0
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OFFSET
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1,8
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COMMENTS
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Compare g.f. to the Lambert series identity: Sum_{n>=1} A210826(n)*x^n/(1-x^n) = Sum_{n>=1} x^(n^3), where A210826(n) = kronecker(-3,d(n)) and d(n) is the number of divisors of n.
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LINKS
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Table of n, a(n) for n=1..74.
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FORMULA
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G.f.: Sum_{n>=1} A210826(n)*Fibonacci(n)*x^n/(1 - Lucas(n)*x^n + (-1)^n*x^(2*n)) = Sum_{n>=1} Fibonacci(n^3)*x^(n^3).
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EXAMPLE
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G.f.: A(x) = x + 21*x^8 + 196418*x^27 + 10610209857723*x^64 + 59425114757512643212875125*x^125 +...
where A(x) = x/(1-x-x^2) + (-1)*1*x^2/(1-3*x^2+x^4) + (-1)*2*x^3/(1-4*x^3-x^6) + (0)*3*x^4/(1-7*x^4+x^8) + (-1)*5*x^5/(1-11*x^5-x^10) + (+1)*8*x^6/(1-18*x^6+x^12)+ (-1)*13*x^7/(1-29*x^7-x^14) + (+1)*21*x^8/(1-47*x^8+x^16) + (0)*34*x^9/(1-76*x^9-x^18) + (+1)*55*x^10/(1-123*x^10+x^20)+ (-1)*89*x^11/(1-199*x^11-x^22) +...+ A210826(n)*Fibonacci(n)*x^n/(1 - Lucas(n)*x^n + (-1)^n*x^(2*n)) +...
The sequence A210826(n) = kronecker(-3, d(n)) begins:
[1,-1,-1,0,-1,1,-1,1,0,1,-1,0,-1,1,1,-1,-1,0,-1,0,1,1,-1,-1,0,1,1,...].
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PROG
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(PARI) {A210826(n)=if(n==0, 0, kronecker(-3, numdiv(n)))}
{Lucas(n)=fibonacci(n-1)+fibonacci(n+1)}
{a(n)=polcoeff(sum(m=1, n, A210826(m)*fibonacci(m)*x^m/(1-Lucas(m)*x^m+(-1)^m*x^(2*m)+x*O(x^n))), n)}
for(n=1, 125, print1(a(n), ", "))
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CROSSREFS
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Cf. A210826, A204060.
Sequence in context: A271976 A182370 A259331 * A167263 A324685 A224114
Adjacent sequences: A209611 A209612 A209613 * A209615 A209616 A209617
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KEYWORD
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nonn
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AUTHOR
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Paul D. Hanna, Apr 07 2012
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STATUS
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approved
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