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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). 0
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

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

Table of n, a(n) for n=0..27.

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

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

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-Fran├žois Alcover, Oct 04 2013 *)

PROG

(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 */

CROSSREFS

Cf. A005773.

Sequence in context: A219233 A211276 A018028 * A101308 A279482 A018029

Adjacent sequences:  A045991 A045992 A045993 * A045995 A045996 A045997

KEYWORD

nonn

AUTHOR

N. J. A. Sloane

STATUS

approved

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Last modified September 18 16:38 EDT 2020. Contains 337170 sequences. (Running on oeis4.)