A366774 Number of 2-distant 3-noncrossing partitions of {1,...,n}.

1, 1, 2, 5, 15, 52, 203, 876, 4115, 20765, 111301, 627821, 3698873, 22623354, 142940629, 929208778, 6194162081, 42223649277, 293640007995, 2079196943605, 14964254850197, 109308213994757, 809340696014733, 6067405789245061, 46008536947670701, 352579939415882813

Offset: 0

Author

Juan B. Gil, Nov 13 2023

Keywords

• nonn

Comments

a(n+1) is the binomial transform of A108304.

Examples

			There are 877 partitions of 7 elements, but a(7)=876 because the partition (1,5)(2,6)(3,7)(4) has a 2-distant 3-crossing.
		

References

• Juan B. Gil and Jordan O. Tirrell, A simple bijection for enhanced, classical, and 2-distant k-noncrossing partitions, Discrete Math. 343 (2020), no. 6, 111705, 5 pp.

Links

• Jean-François Alcover, Table of n, a(n) for n = 0..200
• Juan B. Gil and Jordan O. Tirrell, A simple bijection for enhanced, classical, and 2-distant k-noncrossing partitions, arXiv:1806.09065 [math.CO], 2018-2023.

Crossrefs

Cf. A108304, A366775, A366776.

Programs

Mathematica

b[n_] := b[n] = If[n < 2, 1, (2*(5*n^2 + 12*n - 2)*b[n - 1] + 9*(-n^2 + n + 2)*b[n - 2])/((n + 4)*(n + 5))];
a[n_] := If[n == 0, 1, Sum[Binomial[n - 1, i]*b[i], {i, 0, n - 1}]];
Table[a[n], {n, 0, 200}] (* Jean-François Alcover, Nov 25 2023 *)

Formula

a(n+1) = Sum_{i=0..n} binomial(n,i)*A108304(i).

a(n) ~ 2^(n+1) * 5^(n+7) / (3^(9/2) * Pi * n^7). - Vaclav Kotesovec, Jan 04 2024

Extensions

More terms from Jean-François Alcover, Nov 25 2023