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%I #8 May 08 2018 06:23:10
%S 1,0,36,120,1320,8712,70356,691119,6628050,55398200,528441056,
%T 5607882072,55953959256,559256993400,6033783063160,66852986570260,
%U 743874599106485,8455383000184208,100088596628849400,1202568046655647100,14764362076427728050
%N Number of set partitions of [n] with maximal block length multiplicity equal to seven.
%C At least one block length occurs exactly 7 times, and all block lengths occur at most 7 times.
%H Alois P. Heinz, <a href="/A271736/b271736.txt">Table of n, a(n) for n = 7..592</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, 7)-b(n$2, 6):
%p seq(a(n), n=7..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, 7] - b[n, n, 6];
%t Table[a[n], {n, 7, 30}] (* _Jean-François Alcover_, May 08 2018, after _Alois P. Heinz_ *)
%Y Column k=7 of A271423.
%K nonn
%O 7,3
%A _Alois P. Heinz_, Apr 13 2016
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