login
This site is supported by donations to The OEIS Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
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.

Index entries for reversions of series

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)

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

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent | More pages
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy .

Last modified February 25 22:18 EST 2018. Contains 299662 sequences. (Running on oeis4.)