A390908 Triangle read by rows: T(n,k) is the number of noncrossing path sets on n nodes with k paths and each path having at least two nodes, 0 <= k <= floor(n/2).

1, 0, 0, 1, 0, 3, 0, 8, 2, 0, 20, 15, 0, 48, 75, 5, 0, 112, 308, 63, 0, 256, 1120, 476, 14, 0, 576, 3744, 2772, 252, 0, 1280, 11760, 13680, 2580, 42, 0, 2816, 35200, 60060, 19635, 990, 0, 6144, 101376, 241472, 123255, 12870, 132, 0, 13312, 282880, 906048, 673816, 121550, 3861

Offset

0,6

Comments

A noncrossing path set is a noncrossing forest (A054727) where each tree is restricted to being a path. Paths are constructed using noncrossing line segments between the vertices of a regular n-gon. Isolated vertices are not allowed.

Links

Andrew Howroyd, Table of n, a(n) for n = 0..2600 (rows 0..100)

Formula

G.f.: (1/x) * Series_Reversion( x*(1 - 2*x)^2/(1 - 4*x + (4 + y)*x^2 - y*x^3) ).

Example

Triangle begins:
  1;
  0;
  0,    1;
  0,    3;
  0,    8,     2;
  0,   20,    15;
  0,   48,    75,     5;
  0,  112,   308,    63;
  0,  256,  1120,   476,    14;
  0,  576,  3744,  2772,   252;
  0, 1280, 11760, 13680,  2580,  42;
  0, 2816, 35200, 60060, 19635, 990;
  ...
See examples in A303730.

Prog

(PARI)
T(n)=[Vecrev(p) | p<-Vec(serreverse(x*(1 - 2*x)^2/(1 - 4*x + (4 + y)*x^2 - y*x^3) + O(x^(n+2))))]
{ my(A=T(11)); for(i=1, #A, print(A[i])) }

Crossrefs

Row sums are A303730.

Columns 0..3 are A000007, A001792(n-2), A308914, A362786.

Cf. A054727, A390893 (paths may cross each other), A390909 (singleton vertices allowed).

Sequence in context: A197416 A197512 A232272 * A186744 A200507 A390893

Adjacent sequences: A390905 A390906 A390907 * A390909 A390910 A390911

Keyword

nonn,tabf

Author

Andrew Howroyd, Nov 27 2025

Status

approved