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A271739 Number of set partitions of [n] with maximal block length multiplicity equal to ten.

%I A271739 #8 May 08 2018 06:23:43
%S A271739 1,0,66,286,4004,33033,328328,3150576,31286970,316394650,3928974907,
%T A271739 48404715723,526502083107,6475762500130,88834932638892,
%U A271739 1136875206056150,14448572171583550,197345257083676845,2738327374576989195,37603158111513714720,528367079280330690400
%N A271739 Number of set partitions of [n] with maximal block length multiplicity equal to ten.
%C A271739 At least one block length occurs exactly 10 times, and all block lengths occur at most 10 times.
%H A271739 Alois P. Heinz, <a href="/A271739/b271739.txt">Table of n, a(n) for n = 10..584</a>
%H A271739 Wikipedia, <a href="https://en.wikipedia.org/wiki/Partition_of_a_set">Partition of a set</a>
%p A271739 with(combinat):
%p A271739 b:= proc(n, i, k) option remember; `if`(n=0, 1,
%p A271739       `if`(i<1, 0, add(multinomial(n, n-i*j, i$j)
%p A271739         *b(n-i*j, i-1, k)/j!, j=0..min(k, n/i))))
%p A271739     end:
%p A271739 a:= n-> b(n$2, 10)-b(n$2, 9):
%p A271739 seq(a(n), n=10..30);
%t A271739 multinomial[n_, k_List] := n!/Times @@ (k!);
%t A271739 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 A271739 a[n_] := b[n, n, 10] - b[n, n, 9];
%t A271739 Table[a[n], {n, 10, 30}] (* _Jean-François Alcover_, May 08 2018, after _Alois P. Heinz_ *)
%Y A271739 Column k=10 of A271423.
%K A271739 nonn
%O A271739 10,3
%A A271739 _Alois P. Heinz_, Apr 13 2016