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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.
2
0, 1, 7, 73, 1075, 21066, 527122, 16313963, 609352653, 26938878757, 1387465470527, 82169954359252, 5534425340505464, 419977314311140561, 35617039966665620743, 3352008343756176938273, 347915661537105210844323, 39607489635223003610928042
OFFSET
0,3
LINKS
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
KEYWORD
nonn
AUTHOR
Alois P. Heinz, Sep 08 2019
STATUS
approved