A143646 Catalan transform of the 3-Fibonacci sequence A006190.

0, 1, 4, 18, 83, 387, 1815, 8541, 40276, 190182, 898844, 4250780, 20111394, 95181166, 450565602, 2133227418, 10101126723, 47834649675, 226540406571, 1072931019393, 5081776592061, 24069823974879, 114009427284309

Offset: 0

Author

Sergio Falcon, Oct 27 2008

Keywords

• nonn

Links

• Paul Barry, A Catalan Transform and Related Transformations of Integer Sequences, Journal of Integer Sequences, Vol. 8 (2005), Article 05.4.4
• Sergio Falcón and Ángel Plaza, On the Fibonacci k-numbers, Chaos, Solitons & Fractals 2007; 32(5): 1615-24.
• Sergio Falcón and Ángel Plaza, The k-Fibonacci sequence and the Pascal 2-triangle Chaos, Solitons & Fractals 2007; 33(1): 38-49.

Crossrefs

Cf. A006190, A143464.

Programs

Sage

q=50 #change q for more terms
[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

Formula

a(n) = Sum_{k=0..n} A039599(n,k)*A006130(k-1) with A006130(-1) = 0. - Philippe Deléham, Nov 01 2008

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