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A144700 Generalized (3,-1) Catalan numbers 0

%I

%S 1,1,1,1,2,5,11,21,38,71,141,289,591,1195,2410,4897,10051,20763,42996,

%T 89139,185170,385809,806349,1689573,3547152,7459715,15714655,33161821,

%U 70095642,148388521,314562189,667682057,1418942341

%N Generalized (3,-1) Catalan numbers

%C 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)).

%H S. B. Ekhad, M. Yang, <a href="http://sites.math.rutgers.edu/~zeilberg/tokhniot/oMathar1maple12.txt"> Proofs of Linear Recurrences of Coefficients of Certain Algebraic Formal Power Series Conjectured in the On-Line Encyclopedia Of Integer Sequences</a>, (2017)

%F 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)};

%F 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

%Y Cf. A023431.

%K easy,nonn

%O 0,5

%A _Paul Barry_, Sep 19 2008

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Last modified May 16 14:39 EDT 2021. Contains 343949 sequences. (Running on oeis4.)