A325478 Number of colored set partitions of [n] where colors of the elements of subsets are distinct and in increasing order and all colors of an initial interval of the color palette are used.

1, 1, 4, 29, 329, 5252, 110955, 2972769, 97922354, 3872594811, 180459028989, 9759149087646, 604841170643957, 42508077480226893, 3357224252026104140, 295651782273190911233, 28834727303442640011901, 3095877335697619795977036, 363977673792652615285223095

Offset: 0

Author

Alois P. Heinz, Sep 06 2019

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..296

Crossrefs

Row sums of A322670.

Programs

Maple

b:= proc(n, k) option remember; `if`(n=0, 1, add(b(n-j, k)*
      binomial(n-1, j-1)*binomial(k, j), j=1..min(k, n)))
    end:
a:= n-> add(add(b(n, k-i)*(-1)^i*binomial(k, i), i=0..k), k=0..n):
seq(a(n), n=0..23);

Mathematica

b[n_, k_] := b[n, k] = If[n == 0, 1, Sum[b[n - j, k] Binomial[n - 1, j - 1] Binomial[k, j], {j, 1, Min[k, n]}]];
a[n_] := Sum[Sum[b[n, k - i] (-1)^i Binomial[k, i], {i, 0, k}], {k, 0, n}];
a /@ Range[0, 23] (* Jean-François Alcover, Dec 14 2020, after Alois P. Heinz *)