A200473 - OEIS

%I #28 Mar 24 2017 03:49:50

%S 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,

%T 1,462,5775,15400,10395,1,1,1,1,1716,135135,1,1,126126,1401400,1,1,

%U 6435,2627625,2027025,1,1,1,1,24310,2858856,190590400,34459425,1,1

%N 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).

%C This sequence is A200472 with zeros removed.

%H Alois P. Heinz, <a href="/A200473/b200473.txt">Rows n = 1..250, flattened</a>

%H Dennis P. Walsh, <a href="http://frank.mtsu.edu/~dwalsh/GROUPCT3.pdf">Note on assigning n people to k unlabeled groups of equal size</a>

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

%F Sum_{k=1..tau(k)} T(n,k) = A038041(n). - _Alois P. Heinz_, Jul 22 2016

%e T(n,k) begins:

%e 1;

%e 1,      1;

%e 1,      1;

%e 1,      3,       1;

%e 1,      1;

%e 1,     10,      15,       1;

%e 1,      1;

%e 1,     35,     105,       1;

%e 1,    280,       1;

%e 1,    126,     945,       1;

%e 1,      1;

%e 1,    462,    5775,   15400, 10395,   1;

%e 1,      1;

%e 1,   1716,  135135,       1;

%e 1, 126126, 1401400,       1;

%e 1,   6435, 2627625, 2027025,     1;

%p with(numtheory):

%p S:= n-> sort([divisors(n)[]]):

%p T:= (n,k)-> n!/(S(n)[k])!/((n/(S(n)[k]))!)^(S(n)[k]):

%p seq(seq(T(n, k), k=1..tau(n)), n=1..10);

%t row[n_] := (n!/#!)/(n/#)!^#& /@ Divisors[n];

%t Table[row[n], {n, 1, 20}] // Flatten (* _Jean-François Alcover_, Mar 24 2017 *)

%Y Cf. A200472, A000005 (row lengths).

%Y Cf. A038041 (row sums).

%K nonn,tabf

%O 1,7

%A _Dennis P. Walsh_, Nov 18 2011