proposed

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proposed

multinomial[n_, k_List] := n!/Times @@ (k!);

b[n_, i_, t_] := b[n, i, t] = If[n == 0, {1, 0}, If[i < 1, {0, 0},

Sum[Function[p, p + If[t > 0 && t - j < 1, {0, p[[1]]*i}, {0, 0}]][

multinomial[n, Append[Table[i, {j}], n - i*j]]/j!*

b[n - i*j, Min[n - i*j, i - 1], Max[0, t - j]]], {j, 0, n/i}]]];

a[n_] := b[n, n, 4][[2]];

a /@ Range[4, 25] (* Jean-François Alcover, Apr 24 2021, after Alois P. Heinz *)

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Alois P. Heinz, <a href="/A333061/b333061.txt">Table of n, a(n) for n = 4..576</a>

b:= proc(n, i, t) option remember; `if`(n=0, [1, 0], `if`(i<1, 0,

add((p-> p+`if`(t>0 and t-j<1, [0, p[1]*i], 0))(

combinat[multinomial](n, i$j, n-i*j)/j!*

b(n-i*j, min(n-i*j, i-1), max(0, t-j))), j=0..n/i)))

end:

a:= n-> b(n$2, 4)[2]:

seq(a(n), n=4..25);

allocated for Alois P. Heinz

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

1, 11, 81, 512, 3151, 20071, 133853, 924320, 6551293, 47529561, 354259153, 2725545695, 21741995463, 180198265559, 1551865576121, 13865702570254, 128238585735637, 1224733005946425, 12053244176971825, 122035994844818345, 1269623551116437475, 13561114665253219451

4,2

Wikipedia, <a href="https://en.wikipedia.org/wiki/Partition_of_a_set">Partition of a set</a>

Column k=4 of A319375.

allocated

nonn

Alois P. Heinz, Mar 06 2020

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allocated for Alois P. Heinz

allocated

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