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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.
2
1, 1, 4, 29, 329, 5252, 110955, 2972769, 97922354, 3872594811, 180459028989, 9759149087646, 604841170643957, 42508077480226893, 3357224252026104140, 295651782273190911233, 28834727303442640011901, 3095877335697619795977036, 363977673792652615285223095
OFFSET
0,3
LINKS
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 *)
CROSSREFS
Row sums of A322670.
Sequence in context: A295237 A028853 A118795 * A099700 A305636 A276728
KEYWORD
nonn
AUTHOR
Alois P. Heinz, Sep 06 2019
STATUS
approved