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%I #5 Mar 03 2020 12:53:47
%S 1,13,71,396,1877,9199,47371,253108,1420475,8598976,55100124,
%T 369764734,2614650820,19473708445,151503397725,1226996194292,
%U 10339319950504,90530421514787,821670728202320,7714779905351852,74815825933883534,748526174347790448,7717807072187843156
%N Number of entries in the third blocks of all set partitions of [n] when blocks are ordered by increasing lengths.
%H Alois P. Heinz, <a href="/A332943/b332943.txt">Table of n, a(n) for n = 3..576</a>
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Partition_of_a_set">Partition of a set</a>
%p b:= proc(n, i, t) option remember; `if`(n=0, [1, 0], `if`(i>n, 0,
%p add((p-> p+`if`(t>0 and t-j<1, [0, p[1]*i], 0))(b(n-i*j, i+1,
%p max(0, t-j))/j!*combinat[multinomial](n, i$j, n-i*j)), j=0..n/i)))
%p end:
%p a:= n-> b(n, 1, 3)[2]:
%p seq(a(n), n=3..25);
%Y Column k=3 of A319298.
%K nonn
%O 3,2
%A _Alois P. Heinz_, Mar 03 2020
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