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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).
3
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.
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
KEYWORD
nonn,tabf
AUTHOR
Dennis P. Walsh, Nov 18 2011
STATUS
approved