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A333062 Number of entries in the fifth blocks of all set partitions of [n] when blocks are ordered by decreasing lengths.

%I A333062 #8 Apr 24 2021 20:45:09
%S A333062 1,16,162,1345,10096,72973,531015,3984762,30987321,248303940,
%T A333062 2036778980,17044330217,145588640408,1272940217747,11434350878640,
%U A333062 105849240653792,1011701166471075,9987958951272492,101765834737586068,1068365602976497915,11534318293877771406
%N A333062 Number of entries in the fifth blocks of all set partitions of [n] when blocks are ordered by decreasing lengths.
%H A333062 Alois P. Heinz, <a href="/A333062/b333062.txt">Table of n, a(n) for n = 5..576</a>
%H A333062 Wikipedia, <a href="https://en.wikipedia.org/wiki/Partition_of_a_set">Partition of a set</a>
%p A333062 b:= proc(n, i, t) option remember; `if`(n=0, [1, 0], `if`(i<1, 0,
%p A333062       add((p-> p+`if`(t>0 and t-j<1, [0, p[1]*i], 0))(
%p A333062        combinat[multinomial](n, i$j, n-i*j)/j!*
%p A333062       b(n-i*j, min(n-i*j, i-1), max(0, t-j))), j=0..n/i)))
%p A333062     end:
%p A333062 a:= n-> b(n$2, 5)[2]:
%p A333062 seq(a(n), n=5..25);
%t A333062 multinomial[n_, k_List] := n!/Times @@ (k!);
%t A333062 b[n_, i_, t_] := b[n, i, t] = If[n == 0, {1, 0}, If[i < 1, {0, 0}, Sum[
%t A333062      Function[p, p + If[t > 0 && t - j < 1, {0, p[[1]]*i}, {0, 0}]][
%t A333062      multinomial[n, Append[Table[i, {j}], n - i*j]]/j!*
%t A333062      b[n - i*j, Min[n - i*j, i - 1], Max[0, t - j]]], {j, 0, n/i}]]];
%t A333062 a[n_] := b[n, n, 5][[2]];
%t A333062 a /@ Range[5, 25] (* _Jean-François Alcover_, Apr 24 2021, after _Alois P. Heinz_ *)
%Y A333062 Column k=5 of A319375.
%K A333062 nonn
%O A333062 5,2
%A A333062 _Alois P. Heinz_, Mar 06 2020