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

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

Author

N. J. A. Sloane, May 24 1994

Keywords

• sign

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

Examples

			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 + ...
		

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

Crossrefs

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

Programs

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 *)

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 */

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

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

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

Extensions

Extended and signs added by Olivier Gérard

Second formula adapted to offset by Vaclav Kotesovec, Apr 25 2015