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 A007440 Reversion of g.f. for Fibonacci numbers 1,1,2,3,5,... (Formerly M0413) 11
 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; text; internal format)
 OFFSET 1,4 COMMENTS Binomial transform of A104565 (reversion of Pell numbers). - Paul Barry, Mar 15 2005 From Paul Barry, 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) 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). - Paul Barry, Jan 13 2009 REFERENCES N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence). LINKS Vincenzo Librandi, Table of n, a(n) for n = 1..300 P. Barry, Generalized Catalan Numbers, Hankel Transforms and Somos-4 Sequences , J. Int. Seq. 13 (2010) #10.7.2. P. Barry, On the Central Coefficients of Bell Matrices, J. Int. Seq. 14 (2011) # 11.4.3, page 7. FORMULA (n + 3)*a(n + 2) = -(2*n + 3)*a(n + 1) - 5*n*a(n), a(1) = 1, a(2) = -1. G.f.: A(x) = (-1-x+sqrt(1+2*x+5*x^2))/(2*x). a(n) = sum{k=0..floor(n/2), binomial(n, 2k)*C(k)*(-1)^(n-k)}, where C(n) is A000108. - Paul Barry, 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). - Mark van Hoeij, 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. - Vladimir Kruchinin, Sep 21 2010 G.f.: (G(0)-x-1)/(x^2)=1/G(0) where G(k) = 1 + x + x^2/G(k+1) ; (continued fraction). - Sergei N. Gladkovskii, Dec 25 2011 From Peter Bala, Jun 23 2015: (Start) Lucas(n) = [x^n] (x/A(x))^n for n >= 1. -1/A(-x) = 1/x - 1 + x + x^2 - 2*x^4 - 3*x^5 + x^6 + 11*x^7 + 15*x^8 - 13*x^9 + ... is the Laurent series generating function for A214649. (End) a(n) = (-1)^n*hypergeom([1/2 - n/2, -n/2], [2], -4). - Peter Luschny, Mar 19 2018 MAPLE a := n -> (-1)^n*hypergeom([1/2 - n/2, -n/2], [2], -4): seq(simplify(a(n)), n=0..35); # Peter Luschny, Mar 19 2018 MATHEMATICA a[1] = 1; a[2] = -1; a[n_] := a[n] = (-5*(n-2)*a[n-2] + (1-2*n)*a[n-1])/(n+1); Array[a, 36] (* Jean-François Alcover, Apr 18 2014 *) Rest[CoefficientList[Series[(-1-x+Sqrt[1+2*x+5*x^2])/(2*x), {x, 0, 20}], x]] (* Vaclav Kotesovec, Apr 25 2015 *) PROG (PARI) a(n)=polcoeff((-1-x+sqrt(1+2*x+5*x^2+x^2*O(x^n)))/(2*x), n) (PARI) Vec(serreverse(x/(1-x-x^2)+O(x^66))) /* Joerg Arndt, Aug 19 2012 */ (Sage) def A007440_list(len):     T = [0]*(len+1); T[1] = 1; R = [1]     for n in (1..len-1):         a, b, c = 1, 0, 0         for k in range(n, -1, -1):             r = a - b - c             if k < n : T[k+2] = u;             a, b, c = T[k-1], a, b             u = r         T[1] = u; R.append(u)     return R A007440_list(36) # Peter Luschny, Nov 01 2012 CROSSREFS Cf. A000045, A000032, A214649, A291535. Sequence in context: A074307 A163486 A214649 * A100223 A174017 A178081 Adjacent sequences:  A007437 A007438 A007439 * A007441 A007442 A007443 KEYWORD sign AUTHOR N. J. A. Sloane, May 24 1994 EXTENSIONS Extended and signs added by Olivier Gérard Second formula adapted to offset by Vaclav Kotesovec, Apr 25 2015 STATUS approved

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Last modified August 21 14:53 EDT 2018. Contains 313954 sequences. (Running on oeis4.)