A390895 Expansion of e.g.f.: exp(x*(exp(2*x) + 3)/4).

1, 1, 2, 7, 30, 146, 814, 5160, 36412, 281188, 2353296, 21207308, 204592912, 2101390032, 22870108360, 262698581176, 3174062060880, 40221073046288, 533111009413024, 7373468088607152, 106191106069698976, 1589392617825708256, 24679883046981058656, 396946646811220272704

Offset

0,3

Comments

a(n) is the number of sets of noncrossing paths that cover n nodes arranged in a circle with one node paths allowed. 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.

Also, a(n) is the number of sets of noncrossing paths whose nodes are a subset of n nodes arranged in a circle with one node paths disallowed. In this case, one node paths are disallowed, but not every node needs to be covered. The first definition corresponds to the triangle A390894 and this one to the triangle A390896.

Links

Andrew Howroyd, Table of n, a(n) for n = 0..200

Formula

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

Binomial transform of A354323.

Inverse binomial transform of A390907.

Example

The a(3) = 7 set of paths are: {123}, {132}, {213}, {12, 3}, {13, 2}, {23, 1}, {1, 2, 3}.

Prog

(PARI) seq(n)=Vec(serlaplace(exp(x*(exp(2*x + O(x^n)) + 3)/4)));

Crossrefs

Row sums of A390894 and A390896.

Cf. A354323, A390907.

Sequence in context: A394117 A394129 A196148 * A193464 A166990 A059578

Adjacent sequences: A390892 A390893 A390894 * A390896 A390897 A390898

Keyword

nonn

Author

Andrew Howroyd, Nov 23 2025

Status

approved