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A271734 Number of set partitions of [n] with maximal block length multiplicity equal to five.

%I A271734 #8 May 08 2018 06:22:48
%S A271734 1,0,21,56,504,3717,29337,190674,1460745,12532520,100025926,845104624,
%T A271734 7657043576,69364078980,657324748866,6374275533525,64070264089020,
%U A271734 653567576544498,6979149079277683,74951288500334708,835338959385664426,9373747854520238761
%N A271734 Number of set partitions of [n] with maximal block length multiplicity equal to five.
%C A271734 At least one block length occurs exactly 5 times, and all block lengths occur at most 5 times.
%H A271734 Alois P. Heinz, <a href="/A271734/b271734.txt">Table of n, a(n) for n = 5..603</a>
%H A271734 Wikipedia, <a href="https://en.wikipedia.org/wiki/Partition_of_a_set">Partition of a set</a>
%p A271734 with(combinat):
%p A271734 b:= proc(n, i, k) option remember; `if`(n=0, 1,
%p A271734       `if`(i<1, 0, add(multinomial(n, n-i*j, i$j)
%p A271734         *b(n-i*j, i-1, k)/j!, j=0..min(k, n/i))))
%p A271734     end:
%p A271734 a:= n-> b(n$2, 5)-b(n$2, 4):
%p A271734 seq(a(n), n=5..30);
%t A271734 multinomial[n_, k_List] := n!/Times @@ (k!);
%t A271734 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 A271734 a[n_] := b[n, n, 5] - b[n, n, 4];
%t A271734 Table[a[n], {n, 5, 30}] (* _Jean-François Alcover_, May 08 2018, after _Alois P. Heinz_ *)
%Y A271734 Column k=5 of A271423.
%K A271734 nonn
%O A271734 5,3
%A A271734 _Alois P. Heinz_, Apr 13 2016