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Convoluted convolved Fibonacci numbers G_5^(r).
0

%I #12 Dec 09 2017 10:56:49

%S 5,9,17,25,38,51,70,89,115,141,175,209,252,295,348,401,465,529,605,

%T 681,770,859,962,1065,1183,1301,1435,1569,1720,1871,2040,2209,2397,

%U 2585,2793,3001,3230,3459,3710,3961,4235,4509,4807,5105,5428,5751,6100,6449

%N Convoluted convolved Fibonacci numbers G_5^(r).

%H P. Moree, <a href="http://arXiv.org/abs/math.CO/0311205">Convoluted convolved Fibonacci numbers</a>

%F Conjecture: a(n) = (66-18*(-1)^n+(115-3*(-1)^n)*n+36*n^2+2*n^3)/48. G.f.: -x*(x^5-2*x^4-2*x^3+6*x^2+x-5) / ((x-1)^4*(x+1)^2). - _Colin Barker_, Jul 31 2013

%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(r,5),r=1..65);

%t terms = 48;

%t f[z_] := 1/(1 - z - z^2);

%t a[r_] := SeriesCoefficient[(z/r)*Sum[MoebiusMu[d]*f[z^d]^(r/d), {d, Divisors[r]}], {z, 0, 5}];

%t Array[a, terms] (* _Jean-François Alcover_, Dec 09 2017, from Maple *)

%K nonn

%O 1,1

%A _N. J. A. Sloane_, Dec 05 2003

%E Edited by _Emeric Deutsch_, Mar 06 2004