A178519 Triangle read by rows: T(n,k) is the number of ordered trees with n edges and having k vertices of outdegree 2 that have (two) leaves as their (two) children.

1, 1, 1, 1, 4, 1, 11, 3, 32, 10, 98, 33, 1, 309, 114, 6, 998, 402, 30, 3285, 1439, 137, 1, 10981, 5205, 600, 10, 37178, 18976, 2562, 70, 127227, 69610, 10758, 416, 1, 439369, 256626, 44640, 2250, 15, 1529280, 949974, 183594, 11452, 140, 5359314, 3528725

Offset

0,5

Comments

Row n has 1 + floor(n/3) entries (n >= 3).

Sum of entries in row n is A000108(n) (Catalan numbers).

T(n,0) = A178520(n).

Sum_{k>=0} k*T(n,k) = binomial(2n-5, n-2) = A001700(n-3).

Statistic suggested by Lou Shapiro.

Links

Table of n, a(n) for n=0..47.

Formula

G.f. G=G(t,z) satisfies z*G^2 - (1 - z^3 + tz^3)*G + 1 - z^2 + tz^2 = 0.

Example

T(2,1)=1 because we have \/ .
Triangle starts:
   1;
   1;
   1,  1;
   4,  1;
  11,  3;
  32, 10;
  98, 33,  1;

Maple

eq := z*G^2-(1-z^3+t*z^3)*G+1-z^2+t*z^2: G := RootOf(eq, G): Gser := simplify(series(G, z = 0, 18)): for n from 0 to 15 do P[n] := sort(coeff(Gser, z, n)) end do: 1; 1; 1, 1; for n from 3 to 15 do seq(coeff(P[n], t, j), j = 0 .. floor((1/3)*n)) end do; # yields sequence in triangular form

Crossrefs

Cf. A000108, A178520, A001700.

Sequence in context: A111964 A242351 A124324 * A094503 A113897 A158753

Adjacent sequences: A178516 A178517 A178518 * A178520 A178521 A178522

Keyword

nonn,tabf

Author

Emeric Deutsch, May 31 2010

Status

approved