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A325889
Number of colored set partitions of [2n] where colors of the elements of subsets are in (weakly) increasing order and exactly n colors are used.
2
1, 2, 122, 30470, 19946654, 27291293442, 67940872709600, 280154891124993313, 1787697422835498425966, 16765591042116935170071062, 221912878453525607344964295822, 4012317533096874589918210188528948, 96463460015261984561875523126569759208
OFFSET
0,2
LINKS
FORMULA
a(n) = A321296(2n,n).
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(b(2*n, n-i)*(-1)^i*binomial(n, i), i=0..n):
seq(a(n), n=0..15);
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[b[2n, n - i] (-1)^i Binomial[n, i], {i, 0, n}];
a /@ Range[0, 15] (* Jean-François Alcover, Dec 15 2020, after Alois P. Heinz *)
CROSSREFS
Cf. A321296.
Sequence in context: A074490 A056638 A222873 * A013475 A013471 A370967
KEYWORD
nonn
AUTHOR
Alois P. Heinz, Sep 07 2019
STATUS
approved