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A007440 Reversion of g.f. for Fibonacci numbers 1, 1, 2, 3, 5, ....

%I A007440 M0413 #129 Jul 21 2023 09:16:13
%S A007440 1,-1,0,2,-3,-1,11,-15,-13,77,-86,-144,595,-495,-1520,4810,-2485,
%T A007440 -15675,39560,-6290,-159105,324805,87075,-1592843,2616757,2136539,
%U A007440 -15726114,20247800,32296693,-152909577,145139491,417959049,-1460704685,885536173,4997618808,-13658704994
%N A007440 Reversion of g.f. for Fibonacci numbers 1, 1, 2, 3, 5, ....
%C A007440 Binomial transform of A104565 (reversion of Pell numbers). - _Paul Barry_, Mar 15 2005
%C A007440 From _Paul Barry_, Nov 03 2008: (Start)
%C A007440 Hankel transform of a(n) (starting 0,1,-1,..) is F(n)*(-1)^C(n+1,2).
%C A007440 Hankel transform of a(n+1) is (-1)^C(n+1,2).
%C A007440 Hankel transform of a(n+2) is F(n+2)*(-1)^C(n+2,2).
%C A007440 (End)
%C A007440 The sequence 1,1,-1,0,2,... given by 0^n + Sum_{k=0..floor((n-1)/2)} binomial(n-1,2k)*A000108(k)*(-1)^(n-k-1) has Hankel transform F(n+2)*(-1)^binomial(n+1,2). - _Paul Barry_, Jan 13 2009
%C A007440 Apart from signs, essentially the same as A343773. For odd terms, a(n) = A343773(n-1), while a(n) = -A343773(n-1) if n is even. - _Gennady Eremin_, May 19 2021
%D A007440 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A007440 Gennady Eremin, <a href="/A007440/b007440.txt">Table of n, a(n) for n = 1..800</a> (first 300 terms from Vincenzo Librandi)
%H A007440 M. Abramowitz and I. A. Stegun, eds., <a href="http://www.pdmi.ras.ru/~lowdimma/BSD/abramowitz_and_stegun.pdf">Handbook of Mathematical Functions</a>, National Bureau of Standards Applied Math. Series 55, Tenth Printing, 1972 (p. 16, Reversion of Series 3.6.25).
%H A007440 Paul Barry, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL13/Barry1/barry95r.html">Generalized Catalan Numbers, Hankel Transforms and Somos-4 Sequences </a>, J. Int. Seq. 13 (2010) #10.7.2.
%H A007440 Paul Barry, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL14/Barry3/barry132.html">On the Central Coefficients of Bell Matrices</a>, J. Int. Seq. 14 (2011) # 11.4.3, page 7.
%H A007440 Paul Barry, <a href="https://arxiv.org/abs/2104.01644">Centered polygon numbers, heptagons and nonagons, and the Robbins numbers</a>, arXiv:2104.01644 [math.CO], 2021.
%H A007440 Gennady Eremin, <a href="https://arxiv.org/abs/2108.10676">Walking in the OEIS: From Motzkin numbers to Fibonacci numbers. The "shadows" of Motzkin numbers</a>, arXiv:2108.10676 [math.CO], 2021.
%H A007440 <a href="/index/Res#revert">Index entries for reversions of series</a>
%F A007440 D-finite with recurrence (n+3)*a(n+2) = -(2*n + 3)*a(n+1) - 5*n*a(n), a(1) = 1, a(2) = -1.
%F A007440 G.f.: A(x) = (-1 - x + sqrt(1 + 2*x + 5*x^2))/(2*x).
%F A007440 a(n) = Sum_{k=0..floor(n/2)} binomial(n, 2k)*C(k)*(-1)^(n-k), where C(n) is A000108(n). - _Paul Barry_, May 16 2005
%F A007440 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
%F A007440 a(n) = 2^(-n-1)*Sum_{k=floor((n-1)/2)..n} binomial(k+1,n-k)*5^(n-k)*(-1)^k*C(k), n > 0, where C(k) is A000108. - _Vladimir Kruchinin_, Sep 21 2010
%F A007440 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
%F A007440 From _Peter Bala_, Jun 23 2015: (Start)
%F A007440 Lucas(n) = [x^n] (x/A(x))^n for n >= 1.
%F A007440 -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)
%F A007440 a(n) = (-1)^n*hypergeom([1/2 - n/2, -n/2], [2], -4). - _Peter Luschny_, Mar 19 2018
%F A007440 From _Gennady Eremin_, May 09 2021: (Start)
%F A007440 a(n) = -(-1)^n * A343773(n-1), n > 0.
%F A007440 G.f.: A(x) = x*B(-x), where B(x) is the g.f. of A343773.
%F A007440 Limit_{n->infinity} a(n)/A001006(n) = 0. (End)
%F A007440 G.f. A(x) satisfies A(x) + 1 + x^-1 = 1/A(x). - _Gennady Eremin_, May 29 2021
%e A007440 G.f. = x - x^2 + 2*x^4 - 3*x^5 - x^6 + 11*x^7 - 15*x^8 - 13*x^9 + 77*x^10 - 86*x^11 - 144*x^12 + ...
%p A007440 A007440 := n -> (-1)^(n+1)*hypergeom([1 - n/2, 1/2 -n/2], [2], -4):
%p A007440 seq(simplify(A007440(n)), n=1..35); # _Peter Luschny_, Mar 19 2018, adapted to offset Jul 21 2023
%p A007440 # Using function CompInv from A357588.
%p A007440 CompInv(25, n -> combinat:-fibonacci(n)); # _Peter Luschny_, Oct 07 2022
%t A007440 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 *)
%t A007440 Rest[CoefficientList[Series[(-1-x+Sqrt[1+2*x+5*x^2])/(2*x),{x,0,20}],x]] (* _Vaclav Kotesovec_, Apr 25 2015 *)
%o A007440 (PARI) a(n)=polcoeff((-1-x+sqrt(1+2*x+5*x^2+x^2*O(x^n)))/(2*x),n)
%o A007440 (PARI) Vec(serreverse(x/(1-x-x^2)+O(x^66))) /* _Joerg Arndt_, Aug 19 2012 */
%o A007440 (Sage)
%o A007440 def A007440_list(len):
%o A007440     T = [0]*(len+1); T[1] = 1; R = [1]
%o A007440     for n in (1..len-1):
%o A007440         a,b,c = 1,0,0
%o A007440         for k in range(n,-1,-1):
%o A007440             r = a - b - c
%o A007440             if k < n : T[k+2] = u;
%o A007440             a,b,c = T[k-1],a,b
%o A007440             u = r
%o A007440         T[1] = u; R.append(u)
%o A007440     return R
%o A007440 A007440_list(36) # _Peter Luschny_, Nov 01 2012
%o A007440 (Python)
%o A007440 A007440 = [0, 1, -1]
%o A007440 for n in range(3, 801):
%o A007440     A007440.append( (-(2*n-1)*A007440[-1]
%o A007440       - 5*(n-2)*A007440[-2])//(n+1) )
%o A007440 for n in range(1, 801):
%o A007440     print(n, A007440[n])  # _Gennady Eremin_, May 10 2021
%Y A007440 Cf. A000045, A000032, A214649, A291535, A343773.
%K A007440 sign
%O A007440 1,4
%A A007440 _N. J. A. Sloane_, May 24 1994
%E A007440 Extended and signs added by _Olivier Gérard_
%E A007440 Second formula adapted to offset by _Vaclav Kotesovec_, Apr 25 2015