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A373760 Number of noncrossing partitions of the n-set including a part containing both 1 and n (with n different from 1), with no pair of singletons {i} and {j} that can be merged into {i,j} and leave the partition a noncrossing partition.

%I A373760 #12 Jun 17 2024 08:26:45
%S A373760 0,0,1,2,4,11,30,88,266,825,2613,8408,27421,90422,300987,1010008,
%T A373760 3413027,11604237,39668334,136258178,470060495,1627913941,5657649569,
%U A373760 19725571728,68975054956,241834515725,849993720642,2994348927858,10570741932441,37390372928207,132497284947463
%N A373760 Number of noncrossing partitions of the n-set including a part containing both 1 and n (with n different from 1), with no pair of singletons {i} and {j} that can be merged into {i,j} and leave the partition a noncrossing partition.
%H A373760 Julien Rouyer, <a href="/A373760/b373760.txt">Table of n, a(n) for n = 0..47</a>
%F A373760 With P the generating function of A363448, the generating function Q of (a(n)) is a solution of the system of two equations
%F A373760 P(t)=Q(t)/(1-Q(t))+t/(1-Q(t))^2+1
%F A373760 Q(t)=t/(1-tP(t))-t.
%e A373760 For n=3, the a(3)=2 partitions are {{1,3},{2}} and {{1,2,3}}.
%e A373760 For n=4, the a(4)=4 partitions are {{1,4},{2,3}}, {{1,2,4},{3}}, {{1,3,4},{2}} and {{1,2,3,4}}.
%o A373760 (Sage) t, P, Q = var('t, P, Q')
%o A373760 P = Q / ( 1 - Q ) + t / ( 1 - Q )^2 + 1
%o A373760 solQ=solve([Q == t / (1 - t * P) - t],Q)
%o A373760 q=solQ[1].rhs()
%o A373760 n = 47
%o A373760 DL_Q = (taylor(q, t,0,n)).simplify_full()
%o A373760 Qn = DL_Q.list()
%o A373760 # _Julien Rouyer_, _Wenjie Fang_, and Alain Ninet, Jun 17 2024
%Y A373760 Cf. A363448 (lonely singles partitions), A363449 (marriageable singles partitions), A000108 (noncrossing partitions).
%K A373760 nonn
%O A373760 0,4
%A A373760 _Julien Rouyer_, Jun 17 2024