0,5

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".

a(n) = Sum_{k < 0} A217540(n, k).

a(n) = A000108(n) - A189912(n-1) for n > 0.

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

a(4) = 4 count the Dyck words

[11010100] (()()()) [11011000] (()(()))

[11100100] ((())()) [11101000] ((()())) .

A217539 := proc(n) local k; if n = 0 then 0 else (2*n)!/(n!^2*(n+1)) - add((n-1)!/(((n-1-k)!*iquo(k, 2)!^2)*(iquo(k, 2)+1)), k=0..n-1) fi end: seq(A217539(i), i=0..28);

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 *)

(SageMath)

def A217539(n):

@CachedFunction

def M(n): return (3*(n-1)*M(n-2)+(2*n+1)*M(n-1))/(n+2) if n>1 else 1

@CachedFunction

def catalan(n): return ((4*n-2)*catalan(n-1))/(n+1) if n>0 else 1

return catalan(n) - (n-1)*M(n-2) - M(n-1) if n!=0 else 0

[A217539(i) for i in (0..28)]

Cf. A217540.

Sequence in context: A334827 A202555 A045992 * A046723 A291394 A244616

Adjacent sequences: A217536 A217537 A217538 * A217540 A217541 A217542

Peter Luschny, Oct 22 2012

approved