cp's OEIS Frontend

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

%I A372342 #30 Jun 25 2024 19:22:04
%S A372342 0,1,0,3,4,15,36,105,288,819,2320,6633,19020,54769,158172,458055,
%T A372342 1329552,3867075,11267856,32884953,96111900,281267469,824083260,
%U A372342 2417052267,7096175856,20852160525,61324675776,180488550375,531581605828,1566658748079,4620016882740,13632008884201,40244583972480
%N A372342 Number of noncrossing partitions of [n] that contain exactly one singleton.
%C A372342 Similar to A005043 and linked to A363448.
%H A372342 Julien Rouyer, <a href="/A372342/b372342.txt">Table of n, a(n) for n = 0..299</a>
%H A372342 Julien Rouyer and A. Ninet, <a href="https://hal.science/hal-04281025">Two New Integer Sequences Related to Crossroads and Catalan Numbers</a>, hal-04281025, 2023. See also <a href="https://arxiv.org/abs/2311.07181">arXiv:2311.07181</a> [math.CO], 2023.
%F A372342 a(n) = Sum_{m=1..floor((n+1)/2)} binomial(n, m-1)*binomial(n-m-1, m-2) for n != 1.
%F A372342 a(n) = n*A005043(n-1) for n>=1. - _Ira M. Gessel_, Jun 25 2024
%e A372342 For n=3 the a(3)=3 partitions with exactly one singleton are {{12},{3}}, {{13},{2}}, and {{1},{23}}.
%p A372342 a:= proc(n) option remember; `if`(n<2, n,
%p A372342       2*(n-2)*a(n-1)/(n-1)+3*a(n-2))
%p A372342     end:
%p A372342 seq(a(n), n=0..32);  # _Alois P. Heinz_, Jun 25 2024
%t A372342 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 *)
%t A372342 a[n_] := (-1)^(1 - n) n Hypergeometric2F1[1 - n, 1/2, 2, 4];
%t A372342 Table[a[n], {n, 0, 32}]  (* _Peter Luschny_, Jun 25 2024 *)
%o A372342 (SageMath)
%o A372342 seq = [0,1]
%o A372342 for n in range(2,20):
%o A372342     up = (n+1) // 2
%o A372342     s = 0
%o A372342     for m in range(1,up+1):
%o A372342         s += binomial(n,m-1) * binomial(n-m-1,m-2)
%o A372342     seq.append(s)
%Y A372342 Cf. A005043, A363448.
%K A372342 nonn,easy
%O A372342 0,4
%A A372342 _Julien Rouyer_, Apr 28 2024