OFFSET
0,5
COMMENTS
David Scambler observed that [1, 0, A113682(n-2)] for n>=2 count the Dyck paths of semilength n which satisfy the condition "number of peaks = number of returns + number of hills" and [1, A189912(n-1)] for n>=1 count the paths which satisfy the condition "number of peaks <= number of returns + number of hills".
FORMULA
a(n) = Sum_{k < 0} A217540(n, k).
a(n) = C(n)-(n-1)*M(n-2)-M(n-1) for n > 0; C(n) Catalan, M(n) Motzkin numbers.
Conjecture: 2*(n+1)*(n-3)*a(n) +(-15*n^2+53*n-12)*a(n-1) +(28*n^2-157*n+165)*a(n-2) + 3*(3*n^2+2*n-26)*a(n-3) -18*(2*n-7)*(n-4)*a(n-4)=0. - R. J. Mathar, Nov 11 2012
EXAMPLE
a(4) = 4 count the Dyck words
[11010100] (()()()) [11011000] (()(()))
[11100100] ((())()) [11101000] ((()())) .
MAPLE
MATHEMATICA
MotzkinNumber[n_] := Sum[ Binomial[n+1, k]*Binomial[n+1-k, k-1], {k, 0, Ceiling[(n+1)/2]}]/(n+1); a[0] = a[1] = 0; a[n_] := CatalanNumber[n] - (n-1)*MotzkinNumber[n-2] - MotzkinNumber[n-1]; Table[a[n], {n, 0, 28}] (* Jean-François Alcover, Jun 27 2013, from 3rd formula *)
PROG
CROSSREFS
KEYWORD
nonn
AUTHOR
Peter Luschny, Oct 22 2012
STATUS
approved