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%I #20 Jul 04 2018 05:43:19
%S 1,1,3,7,18,47,125,337,918,2522,6977,19415,54297,152507,429974,
%T 1216297,3450817,9816460,27991422,79989880,229034820,656979399,
%U 1887653560,5431969355,15653355151,45167783715,130491471940,377426429199
%N 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).
%C 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
%H Vladimir Kruchinin, D. V. Kruchinin, <a href="https://arxiv.org/abs/1103.2582">Composita and their properties</a>, arXiv:1103.2582 [math.CO], 2011-2013.
%F 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
%F 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
%t 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 *)
%o (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 */
%Y Cf. A005773.
%K nonn
%O 0,3
%A _N. J. A. Sloane_
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