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A360182 Number of partitions of [n] where each block size occurs at most twice.

%I A360182 #67 Nov 21 2023 03:23:06
%S A360182 1,1,2,4,14,41,152,575,2634,13207,59927,312170,1946870,10547135,
%T A360182 65168469,421552409,3148178034,20138277895,141300123713,1063603633154,
%U A360182 9108280640649,68154636145922,549824347467969,4551458909818969,39948625639349706,406913301246314341
%N A360182 Number of partitions of [n] where each block size occurs at most twice.
%H A360182 Alois P. Heinz, <a href="/A360182/b360182.txt">Table of n, a(n) for n = 0..648</a>
%H A360182 Wikipedia, <a href="https://en.wikipedia.org/wiki/Partition_of_a_set">Partition of a set</a>
%F A360182 a(n) = Sum_{k=0..2} A271423(n,k).
%e A360182 a(0) = 1: (), the empty partition.
%e A360182 a(1) = 1: 1.
%e A360182 a(2) = 2: 12, 1|2.
%e A360182 a(3) = 4: 123, 12|3, 13|2, 1|23.
%e A360182 a(4) = 14: 1234, 123|4, 124|3, 12|34, 12|3|4, 134|2, 13|24, 13|2|4, 14|23, 1|234, 1|23|4, 14|2|3, 1|24|3, 1|2|34.
%e A360182 a(5) = 41: 12345, 1234|5, 1235|4, 123|45, 123|4|5, 1245|3, 124|35, 124|3|5, 125|34, 12|345, 12|34|5, 125|3|4, 12|35|4, 12|3|45, 1345|2, 134|25, 134|2|5, 135|24, 13|245, 13|24|5, 135|2|4, 13|25|4, 13|2|45, 145|23, 14|235, 14|23|5, 15|234, 1|2345, 1|234|5, 15|23|4, 1|235|4, 1|23|45, 145|2|3, 14|25|3, 14|2|35, 15|24|3, 1|245|3, 1|24|35, 15|2|34, 1|25|34, 1|2|345.
%p A360182 b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
%p A360182       add(combinat[multinomial](n, n-i*j, i$j)/j!*
%p A360182         b(n-i*j, i-1), j=0..min(2, n/i))))
%p A360182     end:
%p A360182 a:= n-> b(n$2):
%p A360182 seq(a(n), n=0..25);
%t A360182 multinomial[n_, k_List] := n!/Times @@ (k!);
%t A360182 b[n_, i_] := b[n, i] = If[n == 0, 1, If[i < 1, 0, Sum[multinomial[n, {n - i*j}~Join~ Table[i, {j}]]/j!*b[n - i*j, i - 1], {j, 0, Min[2, n/i]}]]];
%t A360182 a[n_] :=  b[n, n];
%t A360182 Table[a[n], {n, 0, 25}](* _Jean-François Alcover_, Nov 21 2023, after _Alois P. Heinz_ *)
%Y A360182 Cf. A000110, A007837, A114917, A115275, A271423.
%K A360182 nonn
%O A360182 0,3
%A A360182 _Alois P. Heinz_, May 13 2023