A007440 Reversion of g.f. for Fibonacci numbers 1, 1, 2, 3, 5, .... (Formerly M0413)

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

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)} 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

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

References

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Gennady Eremin, Table of n, a(n) for n = 1..800 (first 300 terms from Vincenzo Librandi)

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, Tenth Printing, 1972 (p. 16, Reversion of Series 3.6.25).

Paul Barry, Generalized Catalan Numbers, Hankel Transforms and Somos-4 Sequences , J. Int. Seq. 13 (2010) #10.7.2.

Paul Barry, On the Central Coefficients of Bell Matrices, J. Int. Seq. 14 (2011) # 11.4.3, page 7.

Paul Barry, Centered polygon numbers, heptagons and nonagons, and the Robbins numbers, arXiv:2104.01644 [math.CO], 2021.

Gennady Eremin, Walking in the OEIS: From Motzkin numbers to Fibonacci numbers. The "shadows" of Motzkin numbers, arXiv:2108.10676 [math.CO], 2021.

Index entries for reversions of series

Formula

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.

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(n). - 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_{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

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

From Gennady Eremin, May 09 2021: (Start)

a(n) = -(-1)^n * A343773(n-1), n > 0.

G.f.: A(x) = x*B(-x), where B(x) is the g.f. of A343773.

Limit_{n->infinity} a(n)/A001006(n) = 0. (End)

G.f. A(x) satisfies A(x) + 1 + x^-1 = 1/A(x). - Gennady Eremin, May 29 2021

Example

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 + ...

Maple

A007440 := n -> (-1)^(n+1)*hypergeom([1 - n/2, 1/2 -n/2], [2], -4):
seq(simplify(A007440(n)), n=1..35); # Peter Luschny, Mar 19 2018, adapted to offset Jul 21 2023
# Using function CompInv from A357588.
CompInv(25, n -> combinat:-fibonacci(n)); # Peter Luschny, Oct 07 2022

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
(Python)
A007440 = [0, 1, -1]
for n in range(3, 801):
    A007440.append( (-(2*n-1)*A007440[-1]
      - 5*(n-2)*A007440[-2])//(n+1) )
for n in range(1, 801):
    print(n, A007440[n])  # Gennady Eremin, May 10 2021

Crossrefs

Cf. A000045, A000032, A214649, A291535, A343773.

Sequence in context: A074307 A163486 A214649 * A343773 A100223 A174017

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