A101431 Triangle read by rows: T(n,k) is number of noncrossing trees with n edges and having k nonroot branch nodes.

1, 3, 9, 3, 27, 28, 81, 174, 18, 243, 900, 285, 729, 4185, 2703, 135, 2187, 18144, 19908, 3024, 6561, 74844, 125496, 38640, 1134, 19683, 297432, 711018, 369696, 32886, 59049, 1148175, 3725190, 2943090, 528930, 10206, 177147, 4330260, 18379548, 20588040, 6228585, 363528

Offset

1,2

Comments

Row n contains ceiling(n/2) terms.

Row sums yield the ternary numbers (A001764).

The average number of nonroot branch nodes over all noncrossing trees with n edges is 7n(n-1)(n-2)/(3(3n-1)(3n-2)) ~ 7n/27 (see A045737).

Links

Andrew Howroyd, Table of n, a(n) for n = 1..930

P. Flajolet and M. Noy, Analytic combinatorics of non-crossing configurations, Discrete Math., 204, 203-229, 1999.

M. Noy, Enumeration of noncrossing trees on a circle, Discrete Math., 180, 301-313, 1998.

Formula

T(n, k) = (1/n)binomial(n, k)*Sum_{i=0..min(k, n-1-2k)} 3^(n-1-k-2i)*binomial(k, i)binomial(n-k, k+i+1), 0 <= k <= ceiling(n/2)-1.

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

Example

T(2,0)=3 because /_, _\ and /\ have no nonroot branchnodes.
Triangle begins:
    1;
    3;
    9,    3;
   27,   28;
   81,  174,   18;
  243,  900,  285;
  729, 4185, 2703, 135;
  ...

Maple

T:= proc(n, k) if k=0 then 3^(n-1) else (1/n)*binomial(n, k)*sum(3^(n-1-k-2*i)*binomial(k, i)*binomial(n-k, k+i+1), i=0..min(k, n-1-2*k)) fi end: for n from 1 to 12 do seq(T(n, k), k=0..ceil(n/2)-1) od; # yields sequence in triangular form

Mathematica

t[n_, k_] := (1/n)*Binomial[n, k]*Sum[3^(n - 1 - k - 2i)*Binomial[k, i]*Binomial[n - k, k + i + 1], {i, 0, Min[k, n - 1 - 2k]}]; Table[t[n, k], {n, 1, 12}, {k, 0, Ceiling[n/2] - 1}] // Flatten (* Jean-François Alcover, Jan 21 2013, after Maple *)

Prog

(PARI)
T(n, k)={binomial(n, k)*sum(i=0, min(k, n-1-2*k), 3^(n-1-k-2*i)*binomial(k, i)*binomial(n-k, k+i+1))/n}
for(n=1, 10, for(k=0, ceil(n/2)-1, print1(T(n, k), ", ")); print); \\ Andrew Howroyd, Nov 17 2017

Crossrefs

Cf. A001764, A045737.

Sequence in context: A255583 A339882 A120429 * A120982 A293634 A125143

Adjacent sequences: A101428 A101429 A101430 * A101432 A101433 A101434

Keyword

nonn,tabf

Author

Emeric Deutsch, Jan 17 2005

Status

approved