A108623 G.f. satisfies x = A(x)*(1-A(x))/(1-A(x)-(A(x))^2).

1, 0, -1, -1, 1, 4, 3, -8, -23, -10, 67, 153, 9, -586, -1081, 439, 5249, 7734, -7941, -47501, -53791, 105314, 430119, 343044, -1249799, -3866556, -1730017, 13996097, 34243897, 1947204, -150962373, -296101864, 121857185, 1582561870

Offset: 1

Author

Christian G. Bower, Jun 12 2005

Keywords

• sign

Comments

Row sums of inverse of Riordan array (1/(1-x-x^2), x*(1-x)/(1-x-x^2)) (Cf. A053538). - Paul Barry, Nov 01 2006

Examples

			G.f. = x - x^3 - x^4 + x^5 + 4*x^6 + 3*x^7 - 8*x^8 - 23*x^9 - 10*x^10 + ...
		

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Crossrefs

Except for signs, same as A108624.

Cf. A039978, A105523, A212804.

Programs

Magma

R:=PowerSeriesRing(Rationals(), 41);
Coefficients(R!( (1+x-Sqrt(1-2*x+5*x^2))/(2*(1-x)) )); // G. C. Greubel, Oct 20 2023

Maple

# Using function CompInv from A357588.
CompInv(34, n -> ifelse(n=-1, 1, combinat:-fibonacci(n-2))); # Peter Luschny, Oct 05 2022

Mathematica

CoefficientList[Series[(1+x-Sqrt[1-2*x+5*x^2])/(2*x*(1-x)), {x, 0, 20}], x] (* Vaclav Kotesovec, Feb 08 2014 *)
a[ n_] := SeriesCoefficient[ (1 + x - Sqrt[1 - 2 x + 5 x^2]) / (2 (1 - x)), {x, 0, n}]; (* Michael Somos, May 19 2014 *)
a[ n_] := If[ n < 1, 0, SeriesCoefficient[ InverseSeries[ Series[ (x - x^2) / (1 - x - x^2), {x, 0, n}]], {x, 0, n}]]; (* Michael Somos, May 19 2014 *)

PARI

{a(n) = if( n<0, 0, polcoeff( (1 + x - sqrt(1 - 2*x + 5*x^2 + x^2 * O(x^n))) / (2 * (1 - x)), n))}; /* Michael Somos, May 19 2014 */

PARI

{b(n) = if( n<1, 0, polcoeff( serreverse( (x - x^2) / (1 - x - x^2) + x * O(x^n)), n))}; /* Michael Somos, May 19 2014 */

SageMath

def A108623_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( (1+x-sqrt(1-2*x+5*x^2))/(2*(1-x)) ).list()
a=A108623_list(41); a[1:] # G. C. Greubel, Oct 20 2023

Formula

Binomial transform of A105523. - Paul Barry, Nov 01 2006

G.f.: (1+x-sqrt(1-2*x+5*x^2))/(2*(1-x)). - Paul Barry, Nov 01 2006

Conjecture: n*a(n) +3*(1-n)*a(n-1) +(7*n-18)*a(n-2) +5*(3-n)*a(n-3)=0. - R. J. Mathar, Nov 15 2011

Lim sup_{n->infinity} |a(n)|^(1/n) = sqrt(5). - Vaclav Kotesovec, Feb 08 2014

Series reversion of g.f. of A212804. - Michael Somos, May 19 2014

G.f.: x / (1 - x + x /(1 - x / (1 - x + x / (1 - x / ...)))). - Michael Somos, May 19 2014

0 = a(n)*(25*a(n+1) - 50*a(n+2) + 45*a(n+3) - 20*a(n+4)) + a(n+1)*(-20*a(n+1) + 34*a(n+2) - 44*a(n+3) + 25*a(n+4)) + a(n+2)*(12*a(n+2) - 2*a(n+3) - 6*a(n+4)) + a(n+3)*(a(n+4)) if n>=0. - Michael Somos, May 19 2014