A153008 Catalan number A000108(n) minus Motzkin number A001006(n).

0, 0, 0, 1, 5, 21, 81, 302, 1107, 4027, 14608, 52988, 192501, 701065, 2560806, 9384273, 34504203, 127288011, 471102318, 1749063906, 6513268401, 24323719461, 91081800417, 341929853235, 1286711419527, 4852902998951, 18341683253676

Offset

0,5

Comments

Number of Dyck n-paths with at least one UUU. - David Scambler, Sep 17 2012

Links

Table of n, a(n) for n=0..26.

Formula

a(n) = A000108(n) - A001006(n).

Conjecture: -(n+1)*(n-3)*(n+2)^2*a(n) +3*(n+1)*(2*n^3-n^2-15*n+8)*a(n-1) -(n-1)*(5*n^3-41*n+48)*a(n-2) -6*(n-1)*(n-2)*(2*n-5)*(n+3)*a(n-3)=0, n>=6 - R. J. Mathar, Mar 04 2018

Maple

A001006 := proc(n) (3/2)^(n+2)*add( 3^(-k)*A000108(k-1)*binomial(k, n+2-k), k=1..n+2) ; end:
A153008 := proc(n) A000108(n)-A001006(n) ;
end:
seq(A153008(n), n=0..30) ; # R. J. Mathar, Jan 22 2009

Mathematica

MotzkinNumber = DifferenceRoot[Function[{y, n}, {(-3n-3)*y[n] + (-2n-5)*y[n+1] + (n+4)*y[n+2] == 0, y[0] == 1, y[1] == 1}]];
a[n_] := CatalanNumber[n] - MotzkinNumber[n];
Table[a[n], {n, 0, 26}] (* Jean-François Alcover, Oct 27 2021 *)

Crossrefs

Cf. A000108, A001006.

Sequence in context: A269917 A081038 A273389 * A292878 A346225 A051196

Adjacent sequences: A153005 A153006 A153007 * A153009 A153010 A153011

Keyword

easy,nonn

Author

Omar E. Pol, Dec 20 2008

Status

approved