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A144700
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Generalized (3,-1) Catalan numbers
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0
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1, 1, 1, 1, 2, 5, 11, 21, 38, 71, 141, 289, 591, 1195, 2410, 4897, 10051, 20763, 42996, 89139, 185170, 385809, 806349, 1689573, 3547152, 7459715, 15714655, 33161821, 70095642, 148388521, 314562189, 667682057, 1418942341
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OFFSET
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0,5
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COMMENTS
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Number of lattice paths in the first quadrant from (0,0) to (n,0) using only steps H=(1,0), U=(1,1) and D=(3,-1). Hankel transform has g.f. (1-x^3)/(1+x^4) (A132380 (n+3)).
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LINKS
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Table of n, a(n) for n=0..32.
S. B. Ekhad, M. Yang, Proofs of Linear Recurrences of Coefficients of Certain Algebraic Formal Power Series Conjectured in the On-Line Encyclopedia Of Integer Sequences, (2017)
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FORMULA
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G.f.: (1/(1-x))(c(x^4/(1-x)^3); a(n)=sum{k=0..floor(n/4), C(n-k,3k)*A000108(k)};
Conjecture: (n+4)*a(n) -2*(2n+5)*a(n-1) +6*(n+1)*a(n-2) +2*(1-2*n)*a(n-3) +3*(2-n)*a(n-4) +2*(2*n-7)*a(n-5)=0. - R. J. Mathar, Nov 16 2011
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CROSSREFS
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Cf. A023431.
Sequence in context: A326509 A082775 A023548 * A000785 A049936 A058358
Adjacent sequences: A144697 A144698 A144699 * A144701 A144702 A144703
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KEYWORD
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easy,nonn
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AUTHOR
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Paul Barry, Sep 19 2008
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STATUS
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approved
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