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

0,3

Links

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

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 *)

Crossrefs

Row sums of A321296.

Sequence in context: A143415 A056545 A362111 * A347951 A275787 A009755

Adjacent sequences: A325885 A325886 A325887 * A325889 A325890 A325891

Keyword

nonn

Author

Alois P. Heinz, Sep 07 2019

Status

approved