A390894 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 allowed, 0 <= k <= n.

1, 0, 1, 0, 1, 1, 0, 3, 3, 1, 0, 8, 15, 6, 1, 0, 20, 70, 45, 10, 1, 0, 48, 330, 315, 105, 15, 1, 0, 112, 1596, 2205, 1015, 210, 21, 1, 0, 256, 7840, 15624, 9625, 2660, 378, 28, 1, 0, 576, 38592, 111888, 90972, 32445, 6048, 630, 36, 1, 0, 1280, 188640, 807840, 861420, 389025, 91665, 12390, 990, 45, 1

Offset

0,8

Comments

Each path consists of straight line segments connecting one 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.

Links

Andrew Howroyd, Table of n, a(n) for n = 0..1325 (rows 0..50)

Formula

Column k is the inverse binomial transform of column k of A390897.

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

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

Example

Triangle begins:
  1;
  0,   1;
  0,   1,    1;
  0,   3,    3,     1;
  0,   8,   15,     6,    1;
  0,  20,   70,    45,   10,    1;
  0,  48,  330,   315,  105,   15,   1;
  0, 112, 1596,  2205, 1015,  210,  21,  1;
  0, 256, 7840, 15624, 9625, 2660, 378, 28, 1;
  ...

Prog

(PARI) T(n) = [Vecrev(p) | p<-Vec(serlaplace( exp(y*x*(exp(2*x + O(x^n)) + 3)/4) ))];
{ my(A=T(8)); for(i=1, #A, print(A[i])) }

Crossrefs

Row sums are A390895.

Columns 0..4 are A000007, A001792, A359405, A360021, A360276.

Cf. A390893 (without singleton paths), A390896, A390897 (not necessarily covering).

Sequence in context: A245974 A316989 A390909 * A135009 A092747 A355870

Adjacent sequences: A390891 A390892 A390893 * A390895 A390896 A390897

Keyword

nonn,tabl

Author

Andrew Howroyd, Nov 23 2025

Status

approved