A101276 Triangle read by rows: T(n,k) is the number of ordered trees having n edges and k branches of length 1.

1, 0, 1, 1, 0, 1, 1, 2, 0, 2, 2, 2, 6, 0, 4, 3, 8, 6, 16, 0, 9, 6, 14, 30, 16, 45, 0, 21, 11, 36, 54, 106, 45, 126, 0, 51, 22, 74, 168, 196, 360, 126, 357, 0, 127, 43, 173, 372, 706, 675, 1197, 357, 1016, 0, 323, 87, 378, 981, 1636, 2775, 2268, 3913, 1016, 2907, 0, 835, 176, 860, 2310, 4771, 6660, 10451, 7469, 12644, 2907, 8350, 0, 2188

Offset

0,8

Comments

Row n has n+1 terms (n>=0). Row sums are the Catalan numbers (A000108). Column 0 yields A026418. T(n,n)=A001006(n-1) (n>0) (the Motzkin numbers).

Links

Table of n, a(n) for n=0..77.

Emeric Deutsch, Ordered trees with prescribed root degrees, node degrees and branch lengths, Discrete Math., 282, 2004, 89-94.

J. Riordan, Enumeration of plane trees by branches and endpoints, J. Comb. Theory (A) 19, 1975, 214-222.

Formula

G.f.: G=G(t, z) satisfies z(t+z-tz)G^2-(1-z+tz+z^2-tz^2)G+1-z+tz+z^2-tz^2=0.

Example

T(3,1)=2 because we have the tree with three edges hanging from the root and the tree with one edge hanging from the root at the end of which two edges are hanging.

Maple

G := 1/2/(-z^2+t*z^2-t*z)*(-1+z-t*z-z^2+t*z^2+sqrt(1-3*t^2*z^2-8*t*z^3+6*t^2*z^3+6*z^4*t-3*t^2*z^4-2*t*z-z^2-3*z^4+2*z^3-2*z+4*t*z^2)): Gser:=simplify(series(G, z=0, 13)): P[0]:=1: for n from 1 to 11 do P[n]:=coeff(Gser, z^n) od: for n from 0 to 11 do seq(coeff(t*P[n], t^k), k=1..n+1) od; # yields the sequence in triangular form

Mathematica

m = 12; G[_] = 0;
Do[G[z_] = (1 + t z - t z^2 - z + z^2 + G[z]^2 (t z - t z^2 + z^2))/(1 + t z - t z^2 - z + z^2) + O[z]^m, {m}];
CoefficientList[#, t]& /@ CoefficientList[G[z], z] // Flatten (* Jean-François Alcover, Nov 15 2019 *)

Crossrefs

Cf. A000108, A000106, A026418.

Sequence in context: A291308 A207944 A063088 * A339434 A348325 A103863

Adjacent sequences: A101273 A101274 A101275 * A101277 A101278 A101279

Keyword

nonn,tabl

Author

Emeric Deutsch, Dec 20 2004

Status

approved