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A271738 Number of set partitions of [n] with maximal block length multiplicity equal to nine.

%I A271738 #8 May 08 2018 06:23:36
%S A271738 1,0,55,220,2860,22022,205205,1853280,17381650,200982925,2291851991,
%T A271738 23049864630,262234646310,3319690300850,39333605649855,
%U A271738 464026283957060,5880153732068000,75836425964702975,973764622911909400,12796285021434965050,173456578124336807300
%N A271738 Number of set partitions of [n] with maximal block length multiplicity equal to nine.
%C A271738 At least one block length occurs exactly 9 times, and all block lengths occur at most 9 times.
%H A271738 Alois P. Heinz, <a href="/A271738/b271738.txt">Table of n, a(n) for n = 9..586</a>
%H A271738 Wikipedia, <a href="https://en.wikipedia.org/wiki/Partition_of_a_set">Partition of a set</a>
%p A271738 with(combinat):
%p A271738 b:= proc(n, i, k) option remember; `if`(n=0, 1,
%p A271738       `if`(i<1, 0, add(multinomial(n, n-i*j, i$j)
%p A271738         *b(n-i*j, i-1, k)/j!, j=0..min(k, n/i))))
%p A271738     end:
%p A271738 a:= n-> b(n$2, 9)-b(n$2, 8):
%p A271738 seq(a(n), n=9..30);
%t A271738 multinomial[n_, k_List] := n!/Times @@ (k!);
%t A271738 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 A271738 a[n_] := b[n, n, 9] - b[n, n, 8];
%t A271738 Table[a[n], {n, 9, 30}] (* _Jean-François Alcover_, May 08 2018, after _Alois P. Heinz_ *)
%Y A271738 Column k=9 of A271423.
%K A271738 nonn
%O A271738 9,3
%A A271738 _Alois P. Heinz_, Apr 13 2016