OFFSET
0,4
LINKS
Alois P. Heinz, Table of n, a(n) for n = 0..576
Wikipedia, Partition of a set
FORMULA
E.g.f: Sum_{k>0} ((x^k)/(k!) * exp(Sum_{j=1..k-1} (x^j)/(j!))). - John Tyler Rascoe, Sep 09 2024
EXAMPLE
a(1) = 1: 1.
a(2) = 1: 12.
a(3) = 4: 123, 12|3, 13|2, 1|23.
a(4) = 11: 1234, 123|4, 124|3, 12|3|4, 134|2, 13|2|4, 1|234, 1|23|4, 14|2|3, 1|24|3, 1|2|34.
a(5) = 36: 12345, 1234|5, 1235|4, 123|45, 123|4|5, 1245|3, 124|35, 124|3|5, 125|34, 12|345, 125|3|4, 12|3|4|5, 1345|2, 134|25, 134|2|5, 135|24, 13|245, 135|2|4, 13|2|4|5, 145|23, 14|235, 15|234, 1|2345, 1|234|5, 1|235|4, 1|23|4|5, 145|2|3, 14|2|3|5, 1|245|3, 1|24|3|5, 1|2|345, 1|2|34|5, 15|2|3|4, 1|25|3|4, 1|2|35|4, 1|2|3|45.
MAPLE
b:= proc(n, m, t) option remember; `if`(n=0, `if`(t=1, 1, 0),
add(binomial(n-1, j-1)*b(n-j, max(j, m),
`if`(j>m, 1, `if`(j=m, t+1, t))), j=1..n))
end:
a:= n-> b(n, 0$2):
seq(a(n), n=0..25);
PROG
(PARI)
A_x(N) = {my(x='x+O('x^N), f=x+sum(k=2, N, (x^k)/(k!)*exp(sum(j=1, k-1, (x^j)/(j!))))); concat([0], Vec(serlaplace(f)))}
A_x(30) \\ John Tyler Rascoe, Sep 09 2024
CROSSREFS
KEYWORD
nonn
AUTHOR
Alois P. Heinz, May 11 2024
STATUS
approved