A126223 -id:A126223 - cp's OEIS frontend

A126222 Triangle read by rows: T(n,k) is the number of 2-Motzkin paths (i.e., Motzkin paths with blue and red level steps) without red level steps on the x-axis, having length n and k level steps (0 <= k <= n).

1, 0, 1, 1, 0, 1, 0, 4, 0, 1, 2, 0, 11, 0, 1, 0, 15, 0, 26, 0, 1, 5, 0, 69, 0, 57, 0, 1, 0, 56, 0, 252, 0, 120, 0, 1, 14, 0, 364, 0, 804, 0, 247, 0, 1, 0, 210, 0, 1800, 0, 2349, 0, 502, 0, 1, 42, 0, 1770, 0, 7515, 0, 6455, 0, 1013, 0, 1, 0, 792, 0, 11055, 0, 27940, 0, 16962, 0

Offset: 0

Emeric Deutsch, Dec 28 2006

Row sums are the Catalan numbers (A000108).

A166073 appears to be a variant of A126222 where zeros are sorted to the start of each row. - R. J. Mathar, Aug 21 2010

			T(3,1)=4 because we have BUD, UBD, URD and UDB, where U=(1,1), D=(1,-1), B=blue (1,0), R=red (1,0).
Triangle starts:
1
0,1
1,0,1
0,4,0,1
2,0,11,0,1
0,15,0,26,0,1
5,0,69,0,57,0,1
0,56,0,252,0,120,0,1
14,0,364,0,804,0,247,0,1
0,210,0,1800,0,2349,0,502,0,1
42,0,1770,0,7515,0,6455,0,1013,0,1
0,792,0,11055,0,27940,0,16962,0,2036,0,1
132,0,8217,0,57035,0,95458,0,43086,0,4083,0,1
0,3003,0,62062,0,257257,0,305812,0,106587,0,8178,0,1
429,0,37037,0,381381,0,1049685,0,931385,0,258153,0,16369,0,1
0,11440,0,328328,0,2022384,0,3962140,0,2723280,0,614520,0,32752,0,1
1430,0,163592,0,2341976,0,9591764,0,14051660,0,7699800,0,1441928,0,65519,0,1
0,43758,0,1665456,0,14275716,0,41666184,0,47352820,0,21167312,0,3342489,0,131054,0,1
4862,0,712062,0,13527852,0,77161980,0,168567444,0,152915748,0,56818743,0,7667883,0,262125,0,1
...

Alois P. Heinz, Rows n = 0..140, flattened

Cf. A000108, A126223.

G:=(1-sqrt(1-4*z*t-4*z^2+4*z^2*t^2))/2/z/(t+z-t^2*z): Gser:=simplify(series(G,z=0,15)): for n from 0 to 12 do P[n]:=sort(expand(coeff(Gser,z,n))) od: for n from 0 to 12 do seq(coeff(P[n],t,j),j=0..n) od; # yields sequence in triangular form
# second Maple program:
b:= proc(x, y) option remember; `if`(y>x or y<0, 0,
      `if`(x=0, 1, expand(b(x-1, y)*`if`(y=0, 1, 2)*z+
       b(x-1, y+1) +b(x-1, y-1))))
    end:
T:= (n, k)-> coeff(b(n, 0), z, k):
seq(seq(T(n, k), k=0..n), n=0..15);  # Alois P. Heinz, May 20 2014

b[x_, y_] := b[x, y] = If[y>x || y<0, 0, If[x == 0, 1, Expand[b[x-1, y]*If[y == 0, 1, 2]*z + b[x-1, y+1] + b[x-1, y-1]]]]; T[n_, k_] := Coefficient[b[n, 0], z, k]; Table[Table[T[n, k], {k, 0, n}], {n, 0, 15}] // Flatten (* Jean-François Alcover, Feb 19 2015, after Alois P. Heinz *)

T(2n,0) = C(2n,n)/(n+1) (the Catalan numbers; A000108).
Sum_{k=0..n} k*T(n,k) = A126223(n).
G.f.: G = G(t,z) satisfies z(t + z - t^2*z)G^2 - G + 1 = 0.

A245891 Number of labeled increasing unary-binary trees on n nodes whose breadth-first reading word avoids 213 and 321.

Original entry on oeis.org

1, 1, 3, 7, 20, 55, 157, 448

Offset: 1

Manda Riehl, Aug 19 2014

The number of labeled increasing unary-binary trees with an associated permutation avoiding 213 and 321 in the classical sense. The tree's permutation is found by recording the labels in the order in which they appear in a breadth-first search. (Note that a breadth-first search reading word is equivalent to reading the tree labels left to right by levels, starting with the root.)

In some cases, the same breadth-first search reading permutation can be found on differently shaped trees. This sequence gives the number of trees, not the number of permutations.

			When n=4, a(n)=7.  In the Links above we show the seven labeled increasing trees on four nodes whose permutation avoids 213 and 321.

Manda Riehl, The 7 trees when n = 4.

A126223 gives the number of binary trees instead of unary-binary trees. A033638 gives the number of permutations which avoid 213 and 321 that are breadth-first reading words on labeled increasing unary-binary trees.

cp's OEIS Frontend

A126222 Triangle read by rows: T(n,k) is the number of 2-Motzkin paths (i.e., Motzkin paths with blue and red level steps) without red level steps on the x-axis, having length n and k level steps (0 <= k <= n).

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