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A271733 Number of set partitions of [n] with maximal block length multiplicity equal to four.

%I A271733 #8 May 08 2018 06:22:40
%S A271733 1,0,15,35,385,2331,13335,88110,629200,4811235,35992957,276332420,
%T A271733 2325570065,20036259075,171879027000,1583318184855,14476456463826,
%U A271733 139849724906591,1347082690705367,13909222770509990,144001190692525628,1519193757875044900
%N A271733 Number of set partitions of [n] with maximal block length multiplicity equal to four.
%C A271733 At least one block length occurs exactly 4 times, and all block lengths occur at most 4 times.
%H A271733 Alois P. Heinz, <a href="/A271733/b271733.txt">Table of n, a(n) for n = 4..612</a>
%H A271733 Wikipedia, <a href="https://en.wikipedia.org/wiki/Partition_of_a_set">Partition of a set</a>
%p A271733 with(combinat):
%p A271733 b:= proc(n, i, k) option remember; `if`(n=0, 1,
%p A271733       `if`(i<1, 0, add(multinomial(n, n-i*j, i$j)
%p A271733         *b(n-i*j, i-1, k)/j!, j=0..min(k, n/i))))
%p A271733     end:
%p A271733 a:= n-> b(n$2, 4)-b(n$2, 3):
%p A271733 seq(a(n), n=4..30);
%t A271733 multinomial[n_, k_List] := n!/Times @@ (k!);
%t A271733 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 A271733 a[n_] := b[n, n, 4] - b[n, n, 3];
%t A271733 Table[a[n], {n, 4, 30}] (* _Jean-François Alcover_, May 08 2018, after _Alois P. Heinz_ *)
%Y A271733 Column k=4 of A271423.
%K A271733 nonn
%O A271733 4,3
%A A271733 _Alois P. Heinz_, Apr 13 2016