A327843 Number of colored integer partitions of 2n using all colors of an n-set such that a color pattern for part i has i distinct colors in increasing order.

1, 1, 7, 94, 2081, 67390, 2969647, 169299808, 12032189630, 1036485156029, 105880393642170, 12604896326749405, 1724189631362670619, 267831346979691504798, 46782781937811822181581, 9111872329195713764645644, 1964607669245374038857479576

Offset: 0

Author

Alois P. Heinz, Sep 27 2019

Keywords

• nonn

Examples

			a(2) = 7: 2ab2ab, 2ab1a1a, 2ab1a1b, 2ab1b1b 1a1a1a1b, 1a1a1b1b, 1a1b1b1b.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..160

Crossrefs

Cf. A327117.

Programs

Maple

b:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0,
      add(b(n-i*j, min(n-i*j, i-1), k)*binomial(
      binomial(k, i)+j-1, j), j=0..n/i)))
    end:
a:= n-> add(b(2*n$2, i)*(-1)^(n-i)*binomial(n, i), i=0..n):
seq(a(n), n=0..17);

Mathematica

b[n_, i_, k_] := b[n, i, k] = If[n==0, 1, If[i<1, 0, Sum[b[n - i*j, Min[n - i*j, i - 1], k] Binomial[Binomial[k, i] + j - 1, j], {j, 0, n/i}]]];
a[n_] := Sum[b[2n, 2n, i] (-1)^(n-i) Binomial[n, i], {i, 0, n}];
a /@ Range[0, 17] (* Jean-François Alcover, Dec 18 2020, after Alois P. Heinz *)

Formula

a(n) = A327117(2n,n).