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A007440
Reversion of g.f. for Fibonacci numbers 1, 1, 2, 3, 5, ....
32
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, On the Central Coefficients of Bell Matrices, J. Int. Seq. 14 (2011) # 11.4.3, page 7.
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 */
(SageMath)
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
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