A390896 Triangle read by rows: T(n,k) is the number of sets of noncrossing paths of size k whose nodes are a subset of n nodes arranged in a circle with one node paths disallowed, 0 <= k <= floor(n/2).

1, 1, 1, 1, 1, 6, 1, 26, 3, 1, 100, 45, 1, 363, 435, 15, 1, 1274, 3465, 420, 1, 4372, 24794, 7140, 105, 1, 14760, 165942, 95760, 4725, 1, 49205, 1061730, 1116990, 124425, 945, 1, 162382, 6578550, 11891880, 2512125, 62370, 1, 531438, 39796053, 118776900, 43128855, 2349270, 10395

Offset

0,6

Comments

Each path consists of straight line segments connecting two or more nodes on the circle. Each of the n nodes is used by at most one path. Although each path is noncrossing, different paths are allowed to intersect.

Links

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

Formula

Column k is the binomial transform of column k of A390893.

E.g.f.: exp(x + y*x*(exp(2*x) - 1)/4).

E.g.f. of column k: exp(x)*(x*exp(2*x) - x)^k/(4^k*k!).

T(2*n,n) = A001147(n).

Example

Triangle begins:
  1;
  1;
  1,     1;
  1,     6,
  1,    26,       3;
  1,   100,      45,
  1,   363,     435,      15;
  1,  1274,    3465,     420;
  1,  4372,   24794,    7140,    105;
  1, 14760,  165942,   95760,   4725;
  1, 49205, 1061730, 1116990, 124425, 945;
  ...

Prog

(PARI) T(n) = [Vecrev(p) | p<-Vec(serlaplace( exp(x + y*x*(exp(2*x + O(x^n)) - 1)/4) ))];
{ my(A=T(10)); for(i=1, #A, print(A[i])) }

Crossrefs

Row sums are A390895.

Columns 0..3 are A000012, A261064(n-1), A360716, A361284.

Cf. A001147, A390893 (all points covered), A390894, A390897 (singleton paths allowed).

Sequence in context: A281631 A259230 A375550 * A278756 A147327 A145629

Adjacent sequences: A390893 A390894 A390895 * A390897 A390898 A390899

Keyword

nonn,tabf

Author

Andrew Howroyd, Nov 23 2025

Status

approved