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A271732 Number of set partitions of [n] with maximal block length multiplicity equal to three.

%I A271732 #8 May 08 2018 06:22:33
%S A271732 1,0,10,35,245,1036,4984,37990,242330,1387595,10324457,73271562,
%T A271732 550531436,3836993356,32056517432,271603606580,2249464283038,
%U A271732 18909114389770,173349802631034,1639551357457112,15220220305707538,147729311772991971,1423109890697311335
%N A271732 Number of set partitions of [n] with maximal block length multiplicity equal to three.
%C A271732 At least one block length occurs exactly 3 times, and all block lengths occur at most 3 times.
%H A271732 Alois P. Heinz, <a href="/A271732/b271732.txt">Table of n, a(n) for n = 3..626</a>
%H A271732 Wikipedia, <a href="https://en.wikipedia.org/wiki/Partition_of_a_set">Partition of a set</a>
%p A271732 with(combinat):
%p A271732 b:= proc(n, i, k) option remember; `if`(n=0, 1,
%p A271732       `if`(i<1, 0, add(multinomial(n, n-i*j, i$j)
%p A271732         *b(n-i*j, i-1, k)/j!, j=0..min(k, n/i))))
%p A271732     end:
%p A271732 a:= n-> b(n$2, 3)-b(n$2, 2):
%p A271732 seq(a(n), n=3..30);
%t A271732 multinomial[n_, k_List] := n!/Times @@ (k!);
%t A271732 b[n_, i_, k_] := b[n, i, k] = If[n == 0, 1, If[i < 1, 0, Sum[multinomial[n, Join[{n - i*j}, Table[i, j]]]*b[n - i*j, i - 1, k]/j!, {j, 0, Min[k, n/i] }]]];
%t A271732 a[n_] := b[n, n, 3] - b[n, n, 2];
%t A271732 Table[a[n], {n, 3, 30}] (* _Jean-François Alcover_, May 08 2018, after _Alois P. Heinz_ *)
%Y A271732 Column k=3 of A271423.
%K A271732 nonn
%O A271732 3,3
%A A271732 _Alois P. Heinz_, Apr 13 2016