A200473 Irregular triangle read by rows: T(n,k) = number of ways to assign n people to d_k unlabeled groups of equal size (where d_k is the k-th divisor of n).

1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 10, 15, 1, 1, 1, 1, 35, 105, 1, 1, 280, 1, 1, 126, 945, 1, 1, 1, 1, 462, 5775, 15400, 10395, 1, 1, 1, 1, 1716, 135135, 1, 1, 126126, 1401400, 1, 1, 6435, 2627625, 2027025, 1, 1, 1, 1, 24310, 2858856, 190590400, 34459425, 1, 1

Offset: 1

Author

Dennis P. Walsh, Nov 18 2011

Keywords

• nonn
• tabf

Comments

This sequence is A200472 with zeros removed.

Examples

			T(n,k) begins:
1;
1,      1;
1,      1;
1,      3,       1;
1,      1;
1,     10,      15,       1;
1,      1;
1,     35,     105,       1;
1,    280,       1;
1,    126,     945,       1;
1,      1;
1,    462,    5775,   15400, 10395,   1;
1,      1;
1,   1716,  135135,       1;
1, 126126, 1401400,       1;
1,   6435, 2627625, 2027025,     1;
		

Links

• Alois P. Heinz, Rows n = 1..250, flattened
• Dennis P. Walsh, Note on assigning n people to k unlabeled groups of equal size

Crossrefs

Cf. A200472, A000005 (row lengths).

Cf. A038041 (row sums).

Programs

Maple

with(numtheory):
S:= n-> sort([divisors(n)[]]):
T:= (n,k)-> n!/(S(n)[k])!/((n/(S(n)[k]))!)^(S(n)[k]):
seq(seq(T(n, k), k=1..tau(n)), n=1..10);

Mathematica

row[n_] := (n!/#!)/(n/#)!^#& /@ Divisors[n];
Table[row[n], {n, 1, 20}] // Flatten (* Jean-François Alcover, Mar 24 2017 *)

Formula

T(n,k) = (n!/d_k!)/(n/d_k)!^d_k, n>=1, 1<=k<=tau(n), d_k = k-th divisor of n.

Sum_{k=1..tau(k)} T(n,k) = A038041(n). - Alois P. Heinz, Jul 22 2016