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

1,7

Comments

This sequence is A200472 with zeros removed.

Links

Alois P. Heinz, Rows n = 1..250, flattened

Dennis P. Walsh, Note on assigning n people to k unlabeled groups of equal size

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

Example

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;

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 *)

Crossrefs

Cf. A200472, A000005 (row lengths).

Cf. A038041 (row sums).

Sequence in context: A061494 A360161 A141901 * A361948 A180172 A327372

Adjacent sequences: A200470 A200471 A200472 * A200474 A200475 A200476

Keyword

nonn,tabf

Author

Dennis P. Walsh, Nov 18 2011

Status

approved