A101452 Triangle read by rows: T(n,k) is number of noncrossing trees with n edges and having k branches.

1, 2, 1, 4, 4, 4, 8, 12, 24, 11, 16, 32, 96, 88, 41, 32, 80, 320, 440, 410, 146, 64, 192, 960, 1760, 2460, 1752, 564, 128, 448, 2688, 6160, 11480, 12264, 7896, 2199, 256, 1024, 7168, 19712, 45920, 65408, 63168, 35184, 8835, 512, 2304, 18432, 59136, 165312, 294336, 379008, 316656, 159030, 35989

Offset

1,2

Comments

Mirror image of A101449.

T(n,k) = 2^(n-k)*binomial(n-1,k-1)*A030981(k).

Row sums yield the ternary numbers (A001764).

T(n,n) = A030981(n).

The average number of branchnodes over all noncrossing trees with n edges is n(n-1)(19n^2-23n+10)/(3(3n-1)(3n-2)) ~ 19n/27 (see A045738).

Links

Table of n, a(n) for n=1..55.

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) = [2^(n-k)/k]binomial(n-1, k-1)*Sum_{i=1..k} (-2)^(k-i)*binomial(k, i)*binomial(3i, i-1).

G.f.: G(t, z) = 1/(1-F), where F satisfies F = z(t + 2tF^2/(1-F) + tF^2/(1-F)^2 + 2F).

Example

T(2,1)=2 because we have /_ and _\; T(2,2)=1 because we have /\
Triangle begins:
   1;
   2,  1;
   4,  4,  4;
   8, 12, 24, 11;
  16, 32, 96, 88, 41;
  ...

Maple

T:=(n, k)->(2^(n-k)/k)*binomial(n-1, k-1)*sum((-2)^(k-i)*binomial(k, i)*binomial(3*i, i-1), i=1..k):for n from 1 to 10 do seq(T(n, k), k=1..n) od; # yields sequence in triangular form

Mathematica

T[n_, k_] := 2^(n-k)/k Binomial[n-1, k-1] Sum[(-2)^(k-i) Binomial[k, i] Binomial[3i, i-1], {i, 1, k}];
Table[T[n, k], {n, 1, 10}, {k, 1, n}] // Flatten (* Jean-François Alcover, Jul 29 2018 *)

Crossrefs

Cf. A001764, A030981, A045738.

Sequence in context: A219194 A234306 A223012 * A019963 A165417 A193631

Adjacent sequences: A101449 A101450 A101451 * A101453 A101454 A101455

Keyword

nonn,tabl

Author

Emeric Deutsch, Jan 19 2005

Status

approved