A325930 Total number of colors used in all colored set partitions of [n] where colors of the elements of subsets are distinct and in increasing order and the colors span an initial interval of the color palette.

0, 1, 7, 73, 1075, 21066, 527122, 16313963, 609352653, 26938878757, 1387465470527, 82169954359252, 5534425340505464, 419977314311140561, 35617039966665620743, 3352008343756176938273, 347915661537105210844323, 39607489635223003610928042

Offset

0,3

Links

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

Formula

a(n) = Sum_{k=1..n} k * A322670(n,k).

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(k*add(b(n, k-i)*(-1)^i*binomial(k, i), i=0..k), k=0..n):
seq(a(n), n=0..18);

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[k Sum[b[n, k-i] (-1)^i Binomial[k, i], {i, 0, k}], {k, 0, n}];
a /@ Range[0, 18] (* Jean-François Alcover, Dec 15 2020, after Alois P. Heinz *)

Crossrefs

Cf. A322670.

Sequence in context: A048174 A352118 A258379 * A360544 A352123 A364938

Adjacent sequences: A325927 A325928 A325929 * A325931 A325932 A325933

Keyword

nonn

Author

Alois P. Heinz, Sep 08 2019

Status

approved