Revision History for A205973
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#7 by Russ Cox at Fri Mar 30 18:37:34 EDT 2012
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| AUTHOR
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_Paul D. Hanna (pauldhanna(AT)juno.com), _, Feb 04 2012
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Discussion
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| Fri Mar 30
| 18:37
| OEIS Server: https://oeis.org/edit/global/213
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#6 by Paul D. Hanna at Sat Mar 10 10:15:13 EST 2012
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#5 by Paul D. Hanna at Sat Mar 10 10:15:10 EST 2012
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| NAME
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a(n) = Fibonacci(n)*A109041(n) for n>=1, with a(0)=1, where A109041 lists the coefficients in eta(q)^9/eta(q^3)^3.
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| COMMENTS
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Compare the g.f. to the Lambert series of A109041:
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| FORMULA
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G.f.: 1 - 9*Sum_{n>=1} Fibonacci(n)*Kronecker(n,3)*n^2*x^n/(1 - Lucas(n)*x^n + (-1)^n*x^(2*n)).
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| EXAMPLE
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G.f.: A(x) = 1 - 9*x + 27*x^2 - 18*x^3 - 351*x^4 + 1080*x^5 + 216*x^6 +...
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| PROG
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((PARI) {Lucas(n)=fibonacci(n-1)+fibonacci(n+1)}
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| CROSSREFS
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Cf. A109041, A205972, A205974, A203847, A000204 (Lucas).
Cf. A209453 (Pell variant).
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| STATUS
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approved
editing
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#4 by T. D. Noe at Sat Feb 04 21:19:05 EST 2012
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#3 by Paul D. Hanna at Sat Feb 04 19:02:10 EST 2012
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#2 by Paul D. Hanna at Sat Feb 04 19:02:06 EST 2012
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| NAME
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allocated a(n) = Fibonacci(n)*A109041(n) for n>=1, with a(0)=1, where A109041 lists the coefficients Paulin D. Hannaeta(q)^9/eta(q^3)^3.
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| DATA
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1, -9, 27, -18, -351, 1080, 216, -5850, 9639, -306, -35640, 96120, -16848, -356490, 508950, 131760, -1821015, 4139424, 69768, -13621698, 18996120, -4925700, -57383640, 136178064, 21282912, -405810225, 557193870, -1767762, -1859194350, 3887571240, -539161920
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| OFFSET
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0,2
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| COMMENTS
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Compare the g.f. to the Lambert series of A109041:
1 - 9*Sum_{n>=1} Kronecker(n,3)*n^2*x^n/(1-x^n).
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| FORMULA
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G.f.: 1 - 9*Sum_{n>=1} Fibonacci(n)*Kronecker(n,3)*n^2*x^n/(1 - Lucas(n)*x^n + (-1)^n*x^(2*n)).
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| EXAMPLE
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G.f.: A(x) = 1 - 9*x + 27*x^2 - 18*x^3 - 351*x^4 + 1080*x^5 + 216*x^6 +...
where A(x) = 1 - 1*9*x + 1*27*x^2 - 2*9*x^3 - 3*117*x^4 + 5*216*x^5 + 8*27*x^6 - 13*450*x^7 + 21*459*x^8 +...+ Fibonacci(n)*A109041(n)*^n +...
The g.f. is also given by the identity:
A(x) = 1 - 9*( 1*1*x/(1-x-x^2) - 1*4*x^2/(1-3*x^2+x^4) + 3*16*x^4/(1-7*x^4+x^8) - 5*25*x^5/(1-11*x^5-x^10) + 13*49*x^7/(1-29*x^7-x^14) - 21*64*x^8/(1-47*x^8+x^16) +...).
The values of the symbol Kronecker(n,3) repeat [1,-1,0, ...].
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| PROG
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(PARI) {Lucas(n)=fibonacci(n-1)+fibonacci(n+1)}
{a(n)=polcoeff(1 - 9*sum(m=1, n, fibonacci(m)*kronecker(m, 3)*m^2*x^m/(1-Lucas(m)*x^m+(-1)^m*x^(2*m) +x*O(x^n))), n)}
for(n=0, 40, print1(a(n), ", "))
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| CROSSREFS
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Cf. A109041, A205972, A205974, A203847, A000204 (Lucas).
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| KEYWORD
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allocated
sign
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| AUTHOR
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Paul D. Hanna (pauldhanna(AT)juno.com), Feb 04 2012
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| STATUS
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approved
editing
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#1 by Paul D. Hanna at Thu Feb 02 10:14:52 EST 2012
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| NAME
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allocated for Paul D. Hanna
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| KEYWORD
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allocated
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| STATUS
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approved
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