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A271737 Number of set partitions of [n] with maximal block length multiplicity equal to eight.

%I A271737 #8 May 08 2018 06:23:29
%S A271737 1,0,45,165,1980,14157,123123,1042470,11229075,117721175,1085614101,
%T A271737 11354532696,132028149240,1440550986525,15693895739115,
%U A271737 183700174158435,2200557929261230,26295830857171150,323510486572841425,4085513198322259275,52716487743732737925
%N A271737 Number of set partitions of [n] with maximal block length multiplicity equal to eight.
%C A271737 At least one block length occurs exactly 8 times, and all block lengths occur at most 8 times.
%H A271737 Alois P. Heinz, <a href="/A271737/b271737.txt">Table of n, a(n) for n = 8..588</a>
%H A271737 Wikipedia, <a href="https://en.wikipedia.org/wiki/Partition_of_a_set">Partition of a set</a>
%p A271737 with(combinat):
%p A271737 b:= proc(n, i, k) option remember; `if`(n=0, 1,
%p A271737       `if`(i<1, 0, add(multinomial(n, n-i*j, i$j)
%p A271737         *b(n-i*j, i-1, k)/j!, j=0..min(k, n/i))))
%p A271737     end:
%p A271737 a:= n-> b(n$2, 8)-b(n$2, 7):
%p A271737 seq(a(n), n=8..30);
%t A271737 multinomial[n_, k_List] := n!/Times @@ (k!);
%t A271737 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 A271737 a[n_] := b[n, n, 8] - b[n, n, 7];
%t A271737 Table[a[n], {n, 8, 30}] (* _Jean-François Alcover_, May 08 2018, after _Alois P. Heinz_ *)
%Y A271737 Column k=8 of A271423.
%K A271737 nonn
%O A271737 8,3
%A A271737 _Alois P. Heinz_, Apr 13 2016