A108623 - OEIS

%I #22 Oct 20 2023 01:45:29

%S 1,0,-1,-1,1,4,3,-8,-23,-10,67,153,9,-586,-1081,439,5249,7734,-7941,

%T -47501,-53791,105314,430119,343044,-1249799,-3866556,-1730017,

%U 13996097,34243897,1947204,-150962373,-296101864,121857185,1582561870

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

%C 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

%H Vincenzo Librandi, <a href="/A108623/b108623.txt">Table of n, a(n) for n = 1..1000</a>

%F Binomial transform of A105523. - _Paul Barry_, Nov 01 2006

%F G.f.: (1+x-sqrt(1-2*x+5*x^2))/(2*(1-x)). - _Paul Barry_, Nov 01 2006

%F 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

%F Lim sup_{n->infinity} |a(n)|^(1/n) = sqrt(5). - _Vaclav Kotesovec_, Feb 08 2014

%F Series reversion of g.f. of A212804. - _Michael Somos_, May 19 2014

%F G.f.: x / (1 - x + x /(1 - x / (1 - x + x / (1 - x / ...)))). - _Michael Somos_, May 19 2014

%F 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

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

%p # Using function CompInv from A357588.

%p CompInv(34, n -> ifelse(n=-1, 1, combinat:-fibonacci(n-2))); # _Peter Luschny_, Oct 05 2022

%t CoefficientList[Series[(1+x-Sqrt[1-2*x+5*x^2])/(2*x*(1-x)), {x, 0, 20}], x] (* _Vaclav Kotesovec_, Feb 08 2014 *)

%t a[ n_] := SeriesCoefficient[ (1 + x - Sqrt[1 - 2 x + 5 x^2]) / (2 (1 - x)), {x, 0, n}]; (* _Michael Somos_, May 19 2014 *)

%t 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 *)

%o (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 */

%o (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 */

%o (Magma)

%o R<x>:=PowerSeriesRing(Rationals(), 41);

%o Coefficients(R!( (1+x-Sqrt(1-2*x+5*x^2))/(2*(1-x)) )); // _G. C. Greubel_, Oct 20 2023

%o (SageMath)

%o def A108623_list(prec):

%o     P.<x> = PowerSeriesRing(ZZ, prec)

%o     return P( (1+x-sqrt(1-2*x+5*x^2))/(2*(1-x)) ).list()

%o a=A108623_list(41); a[1:] # _G. C. Greubel_, Oct 20 2023

%Y Except for signs, same as A108624.

%Y Cf. A039978, A105523, A212804.

%K sign

%O 1,6

%A _Christian G. Bower_, Jun 12 2005