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A364282 Number of partitions of [n] with distinct block sizes such that each block contains exactly one block size different from its own as an element.

%I A364282 #17 Jul 18 2023 08:57:34
%S A364282 1,0,0,1,1,4,11,24,52,226,969,2281,8960,29898,193202,1075509,3346852,
%T A364282 14280775,75858992,332978617,2839114204,19507400962,75453432614,
%U A364282 383685116089,2030801987312,14025840725149,77948290561659,884660446815877,7273497958681824
%N A364282 Number of partitions of [n] with distinct block sizes such that each block contains exactly one block size different from its own as an element.
%H A364282 Alois P. Heinz, <a href="/A364282/b364282.txt">Table of n, a(n) for n = 0..707</a>
%H A364282 Wikipedia, <a href="https://en.wikipedia.org/wiki/Partition_of_a_set">Partition of a set</a>
%e A364282 a(3) = 1: 13|2.
%e A364282 a(4) = 1: 124|3.
%e A364282 a(5) = 4: 1235|4, 124|35, 125|34, 13|245.
%e A364282 a(6) = 11: 12346|5, 1235|46, 1236|45, 1256|34, 14|2356, 145|2|36, 14|256|3, 146|2|35, 15|246|3, 16|245|3, 156|2|34.
%p A364282 f:= proc(n) option remember; `if`(n<2, 1-n, (n-1)*(f(n-1)+f(n-2))) end:
%p A364282 a:= proc(m) option remember; local b; b:=
%p A364282       proc(n, i, p) option remember; `if`(i*(i+1)/2<n, 0,
%p A364282        `if`(n=0, p!*f(m-p), b(n, i-1, p)+b(n-i, min(n-i, i-1), p-1)/(i-1)!))
%p A364282       end: b(m$3)
%p A364282     end:
%p A364282 seq(a(n), n=0..30);
%Y A364282 Cf. A000110, A000166, A007837, A364207, A363881, A364278, A364283.
%K A364282 nonn
%O A364282 0,6
%A A364282 _Alois P. Heinz_, Jul 17 2023