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A007440
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Reversion of g.f. for Fibonacci numbers 1,1,2,3,5,...
(Formerly M0413)
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7
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1, -1, 0, 2, -3, -1, 11, -15, -13, 77, -86, -144, 595, -495, -1520, 4810, -2485, -15675, 39560, -6290, -159105, 324805, 87075, -1592843, 2616757, 2136539, -15726114, 20247800, 32296693, -152909577, 145139491, 417959049, -1460704685, 885536173, 4997618808, -13658704994
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,4
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COMMENTS
| Binomial transform of A104565 (reversion of Pell numbers). - Paul Barry (pbarry(AT)wit.ie), Mar 15 2005
Contribution from Paul Barry (pbarry(AT)wit.ie), Nov 03 2008: (Start)
Hankel transform of a(n) (starting 0,1,-1,..) is F(n)*(-1)^C(n+1,2).
Hankel transform of a(n+1) is (-1)^C(n+1,2). Hankel transform of a(n+2) is F(n+2)*(-1)^C(n+2,2). (End)
Contribution from Paul Barry (pbarry(AT)wit.ie), Jan 13 2009: (Start)
The sequence 1,1,-1,0,2,... given by 0^n+sum{k=0..floor((n-1)/2), C(n-1,2k)*A000108(k)*(-1)^(n-k-1)}
has Hankel transform F(n+2)*(-1)^C(n+1,2). (End)
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REFERENCES
| N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
| Index entries for reversions of series
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FORMULA
| (n + 3)*a(n + 2) = -(2*n + 3)*a(n + 1) - 5*n*a(n), a(1) = 1, a(2) = -1.
G.f.: (-1-x+sqrt(1+2x+5x^2))/(2x^2).
a(n)=sum{k=0..floor(n/2), binomial(n, 2k)*C(k)*(-1)^(n-k)}, where C(n) is A000108. - Paul Barry (pbarry(AT)wit.ie), May 16 2005
a(n) = (5^((n+1)/2)*LegendreP(n-1,-1/sqrt(5))+5^(n/2)*LegendreP(n,-1/sqrt(5)))/(2*n+2) [From Mark van Hoeij (hoeij(AT)math.fsu.edu), Jul 02 2010]
a(n)=2^(-n-1)*(sum(binomial(k+1,n-k)*5^(n-k)*(-1)^(k+2)*C(k),k,floor((n-1)/2),n)), n>0, where C(k) is A000108. [From Kruchinin Vladimir (kru(AT)ie.tusur.ru), Sep 21 2010]
G.f.: A(x) =(G(0)-x-1)/(x^2)=1/G(0) ; G(k) = 1 + x + x^2/G(k+1) ; (continued fraction). - Sergei N. Gladkovskii, Dec 25 2011
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PROG
| (PARI) a(n)=polcoeff((-1-x+sqrt(1+2*x+5*x^2+x^2*O(x^n)))/(2*x), n)
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CROSSREFS
| Cf. A000045.
Sequence in context: A046222 A074307 A163486 * A100223 A174017 A178081
Adjacent sequences: A007437 A007438 A007439 * A007441 A007442 A007443
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KEYWORD
| sign
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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EXTENSIONS
| Extended and signs added by Olivier Gerard (olivier.gerard(AT)gmail.com)
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