OFFSET
0,3
COMMENTS
a(n+1) is the binomial transform of A108304.
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.
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
EXAMPLE
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.
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 *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Juan B. Gil, Nov 13 2023
EXTENSIONS
More terms from Jean-François Alcover, Nov 25 2023
STATUS
approved