A390893 Triangle read by rows: T(n,k) is the number of sets of noncrossing paths of size k that cover n nodes arranged in a circle with one node paths disallowed, 0 <= k <= floor(n/2).

1, 0, 0, 1, 0, 3, 0, 8, 3, 0, 20, 30, 0, 48, 210, 15, 0, 112, 1260, 315, 0, 256, 6944, 4200, 105, 0, 576, 36288, 45360, 3780, 0, 1280, 182880, 433440, 81900, 945, 0, 2816, 897600, 3825360, 1386000, 51975, 0, 6144, 4316928, 31944000, 20207880, 1663200, 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 exactly one path. Although each path is noncrossing, different paths are allowed to intersect. This differs from the noncrossing path sets studied in A390908.

Links

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

Formula

Column k is the inverse binomial transform of column k of A390896.

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

E.g.f. of column k: x^k*(exp(2*x) - 1)^k/(4^k*k!).

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

Example

Triangle begins:
  1;
  0;
  0,    1;
  0,    3;
  0,    8,      3;
  0,   20,     30;
  0,   48,    210,      15;
  0,  112,   1260,     315;
  0,  256,   6944,    4200,     105;
  0,  576,  36288,   45360,    3780;
  0, 1280, 182880,  433440,   81900,   945;
  0, 2816, 897600, 3825360, 1386000, 51975;
  ...

Prog

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

Crossrefs

Row sums are A354323.

Columns 0..4 are A000007, A001792(n-2), A332426, A359404, A360275.

Cf. A001147, A390894 (with singleton paths), A390896 (not necessarily covering), A390897, A390908 (paths may not cross).

Sequence in context: A390908 A186744 A200507 * A348333 A296042 A361982

Adjacent sequences: A390890 A390891 A390892 * A390894 A390895 A390896

Keyword

nonn,tabf

Author

Andrew Howroyd, Nov 23 2025

Status

approved