A045994 a(0)=1, a(n) = M(n) + Sum_{k=1..n-1} M(k)*a(n-k-1), where M(n) are the Motzkin numbers (A001006).

1, 1, 3, 7, 18, 47, 125, 337, 918, 2522, 6977, 19415, 54297, 152507, 429974, 1216297, 3450817, 9816460, 27991422, 79989880, 229034820, 656979399, 1887653560, 5431969355, 15653355151, 45167783715, 130491471940, 377426429199

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn

Comments

Apparently the number of grand Motzkin paths of length n that avoid DD starting at level 1. That is, avoiding either positive to negative or negative to positive crossings of the x axis. - David Scambler, Jul 04 2013

Links

• Vladimir Kruchinin, D. V. Kruchinin, Composita and their properties, arXiv:1103.2582 [math.CO], 2011-2013.

Crossrefs

Cf. A005773.

Programs

Mathematica

m[0] = 1; m[n_] := m[n] = m[n-1] + Sum[m[k]*m[n-k-2], {k, 0, n-2}]; a[0] = 1; a[n_] := a[n] = m[n] + Sum[m[k]*a[n-k-1], {k, 1, n-1}]; Table[a[n], {n, 0, 27}] (* Jean-François Alcover, Oct 04 2013 *)

Maxima

a(n):=sum(sum(k/i*sum(binomial(i,j)*binomial(j,2*j-i-k),j,0,i)*binomial(n-i+k-1,k-1)*(-1)^(n-i),i,k,n),k,1,n); /* Vladimir Kruchinin, Sep 10 2010 */

Formula

G.f.: 1/(1-x(1+x)*M(x)), where M(x) is the generating function for the Motzkin numbers. a(n) = Sum(Sum(k/i*Sum(binomial(i,j)*binomial(j,2*j-i-k),j,0,i)*binomial(n-i+k-1,k-1)*(-1)^(n-i),i,k,n),k,1,n), n>0. - Vladimir Kruchinin, Sep 10 2010

Conjecture: (n+1)*a(n) + 2*(-2*n-1)*a(n-1) + 2*(-n+3)*a(n-2) + (11*n-19)*a(n-3) + (11*n-27)*a(n-4) + 3*(n-3)*a(n-5) = 0. - R. J. Mathar, Sep 27 2013