A321880 Number of partitions of n into colored blocks of equal parts with colors from a set of size n.

1, 1, 4, 15, 44, 135, 456, 1239, 3424, 8694, 27240, 65846, 171864, 406133, 960848, 2615460, 5998416, 14304089, 32273100, 72271516, 153768520, 385905072, 817485768, 1841794483, 3915726528, 8388036950, 17125197336, 35051814558, 78986793592, 160176485813

Offset: 0

Author

Alois P. Heinz, Aug 27 2019

Keywords

• nonn

Examples

			a(3) = 15: 3a, 3b, 3c, 2a1a, 2a1b, 2a1c, 2b1a, 2b1b, 2b1c, 2c1a, 2c1b, 2c1c, 111a, 111b, 111c.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..1650

Crossrefs

Main diagonal of A321884.

Cf. A000142, A003056, A321878, A325916.

Programs

Maple

b:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0, k*add(
      (t-> b(t, min(t, i-1), k))(n-i*j), j=1..n/i) +b(n, i-1, k)))
    end:
a:= n-> b(n$3):
seq(a(n), n=0..31);

Mathematica

b[n_, i_, k_] := b[n, i, k] = If[n == 0, 1, If[i < 1, 0, Sum[Function[t, b[t, Min[t, i - 1], k]][n - i j], {j, 1, n/i}] k + b[n, i - 1, k]]];
a[n_] := b[n, n, n];
a /@ Range[0, 31] (* Jean-François Alcover, Dec 08 2020, after Alois P. Heinz *)

Formula

a(n) = [x^n] Product_{j=1..n} (1+(n-1)*x^j)/(1-x^j).

a(n) = A321884(n,n).

a(n) = Sum_{i=0..floor((sqrt(1+8*n)-1)/2)} n!/(n-i)! * A321878(n,i).

a(n) = n * A325916(n) for n > 0, a(n) = 1.