A214649 a(-1) = 1 and g.f. A(x) satisfies A(x) - 1/A(x) = 1/x - 1.

1, -1, 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

Offset

-1,6

Comments

The sequence a(n), n >= 1 := 1, 1, 0, -2, -3, 1, 11, 15, -13, -77, ... is the reversion of g.f. for signed Fibonacci numbers A039834 1, -1, 2, -3, 5, -8, 13, -21, 34, -55, ... - Gennady Eremin, May 31 2021

Links

G. C. Greubel, Table of n, a(n) for n = -1..2500

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

Formula

a(n+1) = a(n) - ( a(1) * a(n-1) + a(2) * a(n-2) + ... + a(n-1) * a(1) ) if n > 0.

D-finite recurrence (n + 1) * a(n) - (2*n - 1) * a(n-1) + (5*n - 10) * a(n-2) =0 unless n = 1.

0 = a(n)*(+25*a(n+1) - 25*a(n+2) + 20*a(n+3)) + a(n+1)*(+5*a(n+1) - 6*a(n+2) - 5*a(n+3)) + a(n+2)*(+a(n+2) + a(n+3)) if n>=0. - Michael Somos, Jan 31 2014

G.f.: ( 1 - x + sqrt( 1 - 2*x + 5*x^2 ) ) / (2 * x).

G.f. A(x) = (1/x) * (1 - x + x^2 / (1 - x + x^2 / (1 - x + x^2 / ...))).

a(n) = -(-1)^n * A007440(n) unless n=0 or n=-1. a(n) = A100223(n+1) unless n=0.

a(n) = Sum_{j=0..floor((n+1)/2)} (-1)^(j-1)*binomial(2*j-1, j)*binomial(n-1, 2*j-2))/(2*j-1), n >= 0, a(-1)=1. - Vladimir Kruchinin, May 28 2014

-1/A(-x) is the o.g.f. for A007440. Lucas(n) = [x^n] (-x*A(-x))^n for n >= 1. - Peter Bala, Jun 23 2015

From Gennady Eremin, May 18 2021: (Start)

a(n+1) = A343773(n), n >= 0.

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

G.f. A(x) satisfies A(x) - 1/x + 1 = 1/A(x) = x*B(x), where B(x) is the g.f. of A343773. - Gennady Eremin, May 29 2021

Example

G.f. = x^-1 - 1 + x + x^2 - 2*x^4 - 3*x^5 + x^6 + 11*x^7 + 15*x^8 - 13*x^9 + ...

Maple

A214649 := n -> `if`(n<2, (-1)^(n-1), simplify(hypergeom([(1-n)/2, 1-n/2], [2], -4))): seq(A214649(n), n=-1..34); # Peter Luschny, May 28 2014

Mathematica

CoefficientList[Series[(1-x + Sqrt[1-2*x+5*x^2])/2, {x, 0, 50}], x] (* G. C. Greubel, Aug 10 2018 *)

Prog

(PARI) {a(n) = if( n<-1, 0, polcoeff( ( 1 - x + sqrt( 1 - 2*x + 5*x^2 + x^2 * O(x^n) ) ) / ( 2*x ), n))}
(PARI) {a(n) = local( A ); if( n<-1, 0, A = 1 / x + O(x^0); forstep( k = -1, n, 2, A = 1 / x - 1 + 1 / A ); polcoeff( A, n ) )};
(Maxima)
a(n):=if n=-1 then 1 else sum(((-1)^(j-1)*binomial(2*j-1, j)*binomial(n-1, 2*j-2))/(2*j-1), j, 0, (n+1)/2); /* Vladimir Kruchinin, May 28 2014 */
(Magma) m:=50; R<x>:=PowerSeriesRing(Rationals(), m); Coefficients(R!((1-x + Sqrt(1-2*x+5*x^2))/2)); // G. C. Greubel, Aug 10 2018
(Python)
A214649 = [1, -1, 1]
for n in range(2, 802):
    A214649.append( ((2*n-1)*A214649[-1]
      - 5*(n-2)*A214649[-2])//(n+1) )
for n in range(802):
    print(n-1, A214649[n]) # Gennady Eremin, May 18 2021

Crossrefs

Cf. A007440, A100223, A000032, A001006, A343773, A039834.

Sequence in context: A291647 A074307 A163486 * A007440 A343773 A100223

Adjacent sequences: A214646 A214647 A214648 * A214650 A214651 A214652

Keyword

sign,easy

Author

Michael Somos, Jul 24 2012

Status

approved