A390897 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 allowed, 0 <= k <= n.

1, 1, 1, 1, 3, 1, 1, 9, 6, 1, 1, 30, 33, 10, 1, 1, 105, 185, 85, 15, 1, 1, 369, 1050, 695, 180, 21, 1, 1, 1281, 6027, 5600, 1995, 336, 28, 1, 1, 4380, 35014, 45080, 21315, 4816, 574, 36, 1, 1, 14769, 205326, 364854, 224343, 65331, 10290, 918, 45, 1, 1, 49215, 1209375, 2973270, 2350845, 861777, 171885, 20070, 1395, 55, 1

Offset

0,5

Comments

Each path consists of straight line segments connecting one 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..1325 (rows 0..50)

Formula

Column k is the binomial transform of column k of A390894.

E.g.f.: exp(x + y*x*(exp(2*x) + 3)/4).

E.g.f. of column k: exp(x)*(x*exp(2*x) + 3*x)^k/(4^k*k!).

Example

Triangle begins:
  1;
  1,    1;
  1,    3,     1;
  1,    9,     6,     1;
  1,   30,    33,    10,     1;
  1,  105,   185,    85,    15,    1;
  1,  369,  1050,   695,   180,   21,   1;
  1, 1281,  6027,  5600,  1995,  336,  28,  1;
  1, 4380, 35014, 45080, 21315, 4816, 574, 36, 1;
  ...

Prog

(PARI) T(n) = [Vecrev(p) | p<-Vec(serlaplace( exp(x + y*x*(exp(2*x + O(x^n)) + 3)/4) ))];
{ my(A=T(10)); for(i=1, #A, print(A[i])) }

Crossrefs

Row sums are A390907.

Columns 0..3 are A000012, A360715, A360717, A361285.

Cf. A390893, A390894 (not necessarily covering), A390896 (without singleton paths).

Sequence in context: A171435 A144183 A050153 * A294950 A294609 A204180

Adjacent sequences: A390894 A390895 A390896 * A390898 A390899 A390900

Keyword

nonn,tabl

Author

Andrew Howroyd, Nov 23 2025

Status

approved