A372342 Number of noncrossing partitions of [n] that contain exactly one singleton.

0, 1, 0, 3, 4, 15, 36, 105, 288, 819, 2320, 6633, 19020, 54769, 158172, 458055, 1329552, 3867075, 11267856, 32884953, 96111900, 281267469, 824083260, 2417052267, 7096175856, 20852160525, 61324675776, 180488550375, 531581605828, 1566658748079, 4620016882740, 13632008884201, 40244583972480

Offset: 0

Author

Julien Rouyer, Apr 28 2024

Keywords

• nonn
• easy

Comments

Similar to A005043 and linked to A363448.

Examples

			For n=3 the a(3)=3 partitions with exactly one singleton are {{12},{3}}, {{13},{2}}, and {{1},{23}}.
		

Links

• Julien Rouyer, Table of n, a(n) for n = 0..299
• Julien Rouyer and A. Ninet, Two New Integer Sequences Related to Crossroads and Catalan Numbers, hal-04281025, 2023. See also arXiv:2311.07181 [math.CO], 2023.

Crossrefs

Cf. A005043, A363448.

Programs

Maple

a:= proc(n) option remember; `if`(n<2, n,
      2*(n-2)*a(n-1)/(n-1)+3*a(n-2))
    end:
seq(a(n), n=0..32);  # Alois P. Heinz, Jun 25 2024

Mathematica

a[n_]:=Sum[Binomial[n, m-1]*Binomial[n-m-1, m-2], {m, Floor[(n+1)/2]}]; Array[a,30,0] (* Stefano Spezia, Apr 28 2024 *)
a[n_] := (-1)^(1 - n) n Hypergeometric2F1[1 - n, 1/2, 2, 4];
Table[a[n], {n, 0, 32}]  (* Peter Luschny, Jun 25 2024 *)

SageMath

seq = [0,1]
for n in range(2,20):
    up = (n+1) // 2
    s = 0
    for m in range(1,up+1):
        s += binomial(n,m-1) * binomial(n-m-1,m-2)
    seq.append(s)

Formula

a(n) = Sum_{m=1..floor((n+1)/2)} binomial(n, m-1)*binomial(n-m-1, m-2) for n != 1.

a(n) = n*A005043(n-1) for n>=1. - Ira M. Gessel, Jun 25 2024