A101307 Triangle read by rows: T(n,k) is the number of ordered trees having n edges and k branches of length 2.

1, 1, 1, 3, 2, 7, 6, 1, 18, 18, 6, 47, 59, 24, 2, 129, 188, 96, 16, 362, 605, 369, 90, 4, 1038, 1948, 1395, 436, 45, 3022, 6305, 5164, 1981, 315, 9, 8917, 20460, 18885, 8568, 1830, 126, 26600, 66585, 68352, 35818, 9565, 1071, 21, 80098, 217186, 245497, 145796

Offset

1,4

Comments

Row n has 1+floor(n/2) terms (n>=0).

Row sums are the Catalan numbers (A000108).

Column k=0 yields A101308.

T(2n,n) = A001006(n-1) (n>0) (the Motzkin numbers).

Links

Table of n, a(n) for n=1..52.

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 G=1+P+PG(G-1), where P= z/(1-z)+(t-1)z^2 (for the explicit form see the Maple program).

Example

Triangle begins:
   1;
   1,  1;
   3,  2;
   7,  6,  1;
  18, 18,  6;

Maple

G:=(1+t*z^2-z^2+z^3-t*z^3-sqrt((1+t*z^2-z^2+z^3-t*z^3)*(1-4*z+3*z^2-3*t*z^2-3*z^3+3*t*z^3)))/2/z/(1-z+t*z+z^2-t*z^2): Gserz:=simplify(series(G, z=0, 16)): for n from 1 to 14 do P[n]:=sort(coeff(Gserz, z^n)) od: for n from 1 to 14 do seq(coeff(t*P[n], t^k), k=1..1+floor(n/2)) od; # yields the sequence in triangular form

Crossrefs

Cf. A000108, A001006, A101308.

Sequence in context: A236388 A033318 A093780 * A283997 A096899 A265345

Adjacent sequences: A101304 A101305 A101306 * A101308 A101309 A101310

Keyword

nonn,tabf

Author

Emeric Deutsch, Dec 22 2004

Status

approved