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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.
1
0, 0, 1, 2, 4, 11, 30, 88, 266, 825, 2613, 8408, 27421, 90422, 300987, 1010008, 3413027, 11604237, 39668334, 136258178, 470060495, 1627913941, 5657649569, 19725571728, 68975054956, 241834515725, 849993720642, 2994348927858, 10570741932441, 37390372928207, 132497284947463
OFFSET
0,4
LINKS
FORMULA
With P the generating function of A363448, the generating function Q of (a(n)) is a solution of the system of two equations
P(t)=Q(t)/(1-Q(t))+t/(1-Q(t))^2+1
Q(t)=t/(1-tP(t))-t.
EXAMPLE
For n=3, the a(3)=2 partitions are {{1,3},{2}} and {{1,2,3}}.
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}}.
PROG
(Sage) t, P, Q = var('t, P, Q')
P = Q / ( 1 - Q ) + t / ( 1 - Q )^2 + 1
solQ=solve([Q == t / (1 - t * P) - t], Q)
q=solQ[1].rhs()
n = 47
DL_Q = (taylor(q, t, 0, n)).simplify_full()
Qn = DL_Q.list()
# Julien Rouyer, Wenjie Fang, and Alain Ninet, Jun 17 2024
CROSSREFS
Cf. A363448 (lonely singles partitions), A363449 (marriageable singles partitions), A000108 (noncrossing partitions).
Sequence in context: A193062 A193061 A193060 * A063544 A276145 A148150
KEYWORD
nonn
AUTHOR
Julien Rouyer, Jun 17 2024
STATUS
approved