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A333060 Number of entries in the third blocks of all set partitions of [n] when blocks are ordered by decreasing lengths.

%I A333060 #8 Apr 24 2021 20:44:58
%S A333060 1,7,36,186,1023,5867,34744,211888,1343046,8896185,61801182,449917898,
%T A333060 3425580850,27183592435,224196765392,1917038645772,16963064269986,
%U A333060 155112925687673,1464150720422785,14253033440621462,142967758696293317,1476398153663677539
%N A333060 Number of entries in the third blocks of all set partitions of [n] when blocks are ordered by decreasing lengths.
%H A333060 Alois P. Heinz, <a href="/A333060/b333060.txt">Table of n, a(n) for n = 3..576</a>
%H A333060 Wikipedia, <a href="https://en.wikipedia.org/wiki/Partition_of_a_set">Partition of a set</a>
%p A333060 b:= proc(n, i, t) option remember; `if`(n=0, [1, 0], `if`(i<1, 0,
%p A333060       add((p-> p+`if`(t>0 and t-j<1, [0, p[1]*i], 0))(
%p A333060        combinat[multinomial](n, i$j, n-i*j)/j!*
%p A333060       b(n-i*j, min(n-i*j, i-1), max(0, t-j))), j=0..n/i)))
%p A333060     end:
%p A333060 a:= n-> b(n$2, 3)[2]:
%p A333060 seq(a(n), n=3..24);
%t A333060 multinomial[n_, k_List] := n!/Times @@ (k!);
%t A333060 b[n_, i_, t_] := b[n, i, t] = If[n == 0, {1, 0}, If[i < 1, {0, 0},
%t A333060      Sum[Function[p, p + If[t > 0 && t - j < 1, {0, p[[1]]*i}, {0, 0}]][
%t A333060      multinomial[n, Append[Table[i, {j}], n - i*j]]/j!*
%t A333060      b[n - i*j, Min[n - i*j, i - 1], Max[0, t - j]]], {j, 0, n/i}]]];
%t A333060 a[n_] := b[n, n, 3][[2]];
%t A333060 a /@ Range[3, 24] (* _Jean-François Alcover_, Apr 24 2021, after _Alois P. Heinz_ *)
%Y A333060 Column k=3 of A319375.
%K A333060 nonn
%O A333060 3,2
%A A333060 _Alois P. Heinz_, Mar 06 2020