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

%I A372721 #20 Sep 09 2024 19:57:15
%S A372721 0,1,1,4,11,36,132,596,2809,14608,79448,461748,2844052,18559360,
%T A372721 127712483,925057295,7012810967,55513992168,457415487326,
%U A372721 3913510354554,34702368052772,318406785389976,3018747693634775,29537880351353635,297953826680083794,3095201088676962296
%N A372721 Number of partitions of [n] having exactly one block of maximal size.
%H A372721 Alois P. Heinz, <a href="/A372721/b372721.txt">Table of n, a(n) for n = 0..576</a>
%H A372721 Wikipedia, <a href="https://en.wikipedia.org/wiki/Partition_of_a_set">Partition of a set</a>
%F A372721 E.g.f: Sum_{k>0} ((x^k)/(k!) * exp(Sum_{j=1..k-1} (x^j)/(j!))). - _John Tyler Rascoe_, Sep 09 2024
%e A372721 a(1) = 1: 1.
%e A372721 a(2) = 1: 12.
%e A372721 a(3) = 4: 123, 12|3, 13|2, 1|23.
%e A372721 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.
%e A372721 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.
%p A372721 b:= proc(n, m, t) option remember; `if`(n=0, `if`(t=1, 1, 0),
%p A372721       add(binomial(n-1, j-1)*b(n-j, max(j, m),
%p A372721      `if`(j>m, 1, `if`(j=m, t+1, t))), j=1..n))
%p A372721     end:
%p A372721 a:= n-> b(n, 0$2):
%p A372721 seq(a(n), n=0..25);
%o A372721 (PARI)
%o A372721 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)))}
%o A372721 A_x(30) \\ _John Tyler Rascoe_, Sep 09 2024
%Y A372721 Column k=1 of A372722.
%Y A372721 Cf. A000110, A224219, A372802.
%K A372721 nonn
%O A372721 0,4
%A A372721 _Alois P. Heinz_, May 11 2024