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

0,3

Links

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

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.

Example

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

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 *)

Crossrefs

Main diagonal of A321884.

Cf. A000142, A003056, A321878, A325916.

Sequence in context: A329523 A331317 A259664 * A075673 A062827 A074448

Adjacent sequences: A321877 A321878 A321879 * A321881 A321882 A321883

Keyword

nonn

Author

Alois P. Heinz, Aug 27 2019

Status

approved