A271735 - OEIS

%I #8 May 08 2018 06:22:54

%S 1,0,28,84,840,5082,48279,413127,3093090,26601575,255431176,

%T 2309491548,20998179748,209051155600,2137087555220,21652990622410,

%U 230200208290745,2517313465793819,28104615964752327,320432370881428575,3760667223506993800,45094960570293757695

%N Number of set partitions of [n] with maximal block length multiplicity equal to six.

%C At least one block length occurs exactly 6 times, and all block lengths occur at most 6 times.

%H Alois P. Heinz, <a href="/A271735/b271735.txt">Table of n, a(n) for n = 6..597</a>

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Partition_of_a_set">Partition of a set</a>

%p with(combinat):

%p b:= proc(n, i, k) option remember; `if`(n=0, 1,

%p       `if`(i<1, 0, add(multinomial(n, n-i*j, i$j)

%p         *b(n-i*j, i-1, k)/j!, j=0..min(k, n/i))))

%p     end:

%p a:= n-> b(n$2, 6)-b(n$2, 5):

%p seq(a(n), n=6..30);

%t multinomial[n_, k_List] := n!/Times @@ (k!);

%t 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 a[n_] := b[n, n, 6] - b[n, n, 5];

%t Table[a[n], {n, 6, 30}] (* _Jean-François Alcover_, May 08 2018, after _Alois P. Heinz_ *)

%Y Column k=6 of A271423.

%K nonn

%O 6,3

%A _Alois P. Heinz_, Apr 13 2016