A325888 Number of colored set partitions of [n] where colors of the elements of subsets are in (weakly) increasing order and all colors of an initial interval of the color palette are used.

1, 1, 5, 41, 505, 8597, 191457, 5364837, 183744421, 7521913845, 361544182917, 20109571623693, 1278810836639233, 92032189911692253, 7430335604308535497, 667922294225164998677, 66407623510409091454229, 7260203111052685954056549, 868289612454444952122790277

Offset: 0

Author

Alois P. Heinz, Sep 07 2019

Keywords

• nonn

Links

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

Crossrefs

Row sums of A321296.

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-1, j), j=1..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..21);

Mathematica

b[n_, k_] := b[n, k] = If[n == 0, 1, Sum[b[n - j, k] Binomial[n - 1, j - 1] Binomial[k + j - 1, j], {j, 1, n}]];
a[n_] := Sum[Sum[b[n, k - i] (-1)^i Binomial[k, i], {i, 0, k}], {k, 0, n}];
a /@ Range[0, 21] (* Jean-François Alcover, Dec 15 2020, after Alois P. Heinz *)