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Catalan transform of the 3-Fibonacci sequence A006190.
1

%I #21 Mar 10 2014 15:23:54

%S 0,1,4,18,83,387,1815,8541,40276,190182,898844,4250780,20111394,

%T 95181166,450565602,2133227418,10101126723,47834649675,226540406571,

%U 1072931019393,5081776592061,24069823974879,114009427284309

%N Catalan transform of the 3-Fibonacci sequence A006190.

%H Paul Barry, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL8/Barry/barry84.html">A Catalan Transform and Related Transformations of Integer Sequences</a>, Journal of Integer Sequences, Vol. 8 (2005), Article 05.4.4

%H Sergio Falcón and Ángel Plaza, <a href="http://dx.doi.org/10.1016/j.chaos.2006.09.022">On the Fibonacci k-numbers</a>, Chaos, Solitons & Fractals 2007; 32(5): 1615-24.

%H Sergio Falcón and Ángel Plaza, <a href="http://dx.doi.org/10.1016/j.chaos.2006.10.022">The k-Fibonacci sequence and the Pascal 2-triangle</a> Chaos, Solitons & Fractals 2007; 33(1): 38-49.

%F a(n) = Sum_{k=0..n} A039599(n,k)*A006130(k-1) with A006130(-1) = 0. - _Philippe Deléham_, Nov 01 2008

%F For n>0, a(n) = sum_{k=0..n} (k/n)*C(2n-k-1,n-k)*A006190(k). - _Tom Edgar_, Mar 09 2014

%o (Sage)

%o q=50 #change q for more terms

%o [0]+[sum((k/n)*binomial(2*n-k-1,n-k)*lucas_number1(k,3,-1) for k in [0..n]) for n in [1..q]] # _Tom Edgar_, Mar 09 2014

%Y Cf. A006190, A143464.

%K nonn

%O 0,3

%A _Sergio Falcon_, Oct 27 2008