Number of lattice paths from (0,0) to (n-4,0) that stay weakly in the first quadrant and such that each step is either U=(2,1), D=(2,-1), blue H=(1,0), or red h=(1,0) (n>=4). E.g., a(8)=17 because we have 16 horizontal paths of length 4 with all combinations of blue and red (1,0) steps and, in addition, UD. - Emeric Deutsch, Dec 23 2003
a(n+3) is the number of rooted plane 2-trees with nonempty integer compositions labeling all the nodes, including the root, with total size n >= 0. The total size is the number of edges in the tree plus the sum of the sizes of the integer compositions labeling all the nodes.
Examples: a(3)=0 because there are no elements of size zero; a(4)=1, a(5)=2, a(6)=4 and a(7)=8 because in each case, the elements are trees that consist of the root alone labeled with the compositions of 1, 2, 3 and 4, respectively; a(8)=17 because now we have 17 elements of size 5, the first 16 coming from the root alone labeled with the compositions of 5, plus the 2-tree that consists of the root with two descendants, with each of the three nodes labeled with the composition 1=1. (End)
