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%I #27 Aug 11 2020 04:53:30
%S 1,1,3,5,11,19,37,65,120,210,376,654,1149,1985,3443,5911,10159,17345,
%T 29605,50305,85400,144516,244272,411900,693729,1166209,1958219,
%U 3283145,5498595,9197455,15369373,25655489,42787456,71293590,118695272,197452746,328227725
%N Sign twisted convoluted convolved Fibonacci numbers H_j^(2).
%C Let "a" = the Fibonacci numbers, and "b" = the aerated Fibonacci numbers (1, 0, 1, 0, 2,...). Then A089098 = (1/2) * (a^2 + b), where a^2 = A001629, the Fibonacci numbers convolved with themselves: (1, 2, 5, 10, 20, 38,...).
%H Colin Barker, <a href="/A089098/b089098.txt">Table of n, a(n) for n = 1..1000</a>
%H A. R. Ashrafi, J. Azarija, K. Fathalikhani, S. Klavzar and M. Petkovsek, <a href="http://www.fmf.uni-lj.si/~klavzar/preprints/Fib-Luc-orbits-submit-2014.pdf">Vertex and edge orbits of Fibonacci and Lucas cubes</a>, 2014; See Table 2.
%H P. Moree, <a href="http://arXiv.org/abs/math.CO/0311205">Convoluted convolved Fibonacci numbers</a>, arXiv:math/0311205 [math.CO], 2003.
%H <a href="/index/Rec#order_08">Index entries for linear recurrences with constant coefficients</a>, signature (2,2,-4,-1,0,0,2,1).
%F G.f.: (z/2)[1/(1-z-z^2)^2+1/(1-z^2-z^4)].
%F G.f.: -x*(x-1)^2*(x+1) / ((x^2+x-1)^2*(x^4+x^2-1)). - _Colin Barker_, Jul 23 2015
%p with(numtheory): f := z->-1/(1-z-z^2): m := proc(r,j) d := divisors(r): W := (1/r)*z*sum(mobius(d[i])*f(z^d[i])^(r/d[i]),i=1..nops(d)): Wser := simplify(series(W,z=0,80)): coeff(Wser,z^j) end: seq(m(2,j),j=1..39);
%t (1-x-x^2+x^3)/((1-x-x^2)^2*(1-x^2-x^4)) + O[x]^40 // CoefficientList[#,x]& (* _Jean-François Alcover_, Jan 20 2018 *)
%o (PARI) Vec(-x*(x-1)^2*(x+1)/((x^2+x-1)^2*(x^4+x^2-1)) + O(x^50)) \\ _Colin Barker_, Jul 23 2015
%Y 2nd column of A337009.
%K nonn,easy
%O 1,3
%A _N. J. A. Sloane_, Dec 05 2003
%E Edited by _Emeric Deutsch_, Mar 06 2004
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