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A333061 Number of entries in the fourth blocks of all set partitions of [n] when blocks are ordered by decreasing lengths.

%I A333061 #8 Apr 24 2021 20:45:04
%S A333061 1,11,81,512,3151,20071,133853,924320,6551293,47529561,354259153,
%T A333061 2725545695,21741995463,180198265559,1551865576121,13865702570254,
%U A333061 128238585735637,1224733005946425,12053244176971825,122035994844818345,1269623551116437475,13561114665253219451
%N A333061 Number of entries in the fourth blocks of all set partitions of [n] when blocks are ordered by decreasing lengths.
%H A333061 Alois P. Heinz, <a href="/A333061/b333061.txt">Table of n, a(n) for n = 4..576</a>
%H A333061 Wikipedia, <a href="https://en.wikipedia.org/wiki/Partition_of_a_set">Partition of a set</a>
%p A333061 b:= proc(n, i, t) option remember; `if`(n=0, [1, 0], `if`(i<1, 0,
%p A333061       add((p-> p+`if`(t>0 and t-j<1, [0, p[1]*i], 0))(
%p A333061        combinat[multinomial](n, i$j, n-i*j)/j!*
%p A333061       b(n-i*j, min(n-i*j, i-1), max(0, t-j))), j=0..n/i)))
%p A333061     end:
%p A333061 a:= n-> b(n$2, 4)[2]:
%p A333061 seq(a(n), n=4..25);
%t A333061 multinomial[n_, k_List] := n!/Times @@ (k!);
%t A333061 b[n_, i_, t_] := b[n, i, t] = If[n == 0, {1, 0}, If[i < 1, {0, 0},
%t A333061      Sum[Function[p, p + If[t > 0 && t - j < 1, {0, p[[1]]*i}, {0, 0}]][
%t A333061      multinomial[n, Append[Table[i, {j}], n - i*j]]/j!*
%t A333061      b[n - i*j, Min[n - i*j, i - 1], Max[0, t - j]]], {j, 0, n/i}]]];
%t A333061 a[n_] := b[n, n, 4][[2]];
%t A333061 a /@ Range[4, 25] (* _Jean-François Alcover_, Apr 24 2021, after _Alois P. Heinz_ *)
%Y A333061 Column k=4 of A319375.
%K A333061 nonn
%O A333061 4,2
%A A333061 _Alois P. Heinz_, Mar 06 2020