A372721 Number of partitions of [n] having exactly one block of maximal size.

0, 1, 1, 4, 11, 36, 132, 596, 2809, 14608, 79448, 461748, 2844052, 18559360, 127712483, 925057295, 7012810967, 55513992168, 457415487326, 3913510354554, 34702368052772, 318406785389976, 3018747693634775, 29537880351353635, 297953826680083794, 3095201088676962296

Offset: 0

Author

Alois P. Heinz, May 11 2024

Keywords

• nonn

Examples

			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.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..576
• Wikipedia, Partition of a set

Crossrefs

Column k=1 of A372722.

Cf. A000110, A224219, A372802.

Programs

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);

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

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