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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.
2
1, 7, 36, 186, 1023, 5867, 34744, 211888, 1343046, 8896185, 61801182, 449917898, 3425580850, 27183592435, 224196765392, 1917038645772, 16963064269986, 155112925687673, 1464150720422785, 14253033440621462, 142967758696293317, 1476398153663677539
OFFSET
3,2
LINKS
MAPLE
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, 3)[2]:
seq(a(n), n=3..24);
MATHEMATICA
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, 3][[2]];
a /@ Range[3, 24] (* Jean-François Alcover, Apr 24 2021, after Alois P. Heinz *)
CROSSREFS
Column k=3 of A319375.
Sequence in context: A037538 A037482 A147546 * A331719 A020085 A356076
KEYWORD
nonn
AUTHOR
Alois P. Heinz, Mar 06 2020
STATUS
approved