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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).

%I A200473 #28 Mar 24 2017 03:49:50
%S A200473 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 A200473 1,462,5775,15400,10395,1,1,1,1,1716,135135,1,1,126126,1401400,1,1,
%U A200473 6435,2627625,2027025,1,1,1,1,24310,2858856,190590400,34459425,1,1
%N 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).
%C A200473 This sequence is A200472 with zeros removed.
%H A200473 Alois P. Heinz, <a href="/A200473/b200473.txt">Rows n = 1..250, flattened</a>
%H A200473 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 A200473 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 A200473 Sum_{k=1..tau(k)} T(n,k) = A038041(n). - _Alois P. Heinz_, Jul 22 2016
%e A200473 T(n,k) begins:
%e A200473 1;
%e A200473 1,      1;
%e A200473 1,      1;
%e A200473 1,      3,       1;
%e A200473 1,      1;
%e A200473 1,     10,      15,       1;
%e A200473 1,      1;
%e A200473 1,     35,     105,       1;
%e A200473 1,    280,       1;
%e A200473 1,    126,     945,       1;
%e A200473 1,      1;
%e A200473 1,    462,    5775,   15400, 10395,   1;
%e A200473 1,      1;
%e A200473 1,   1716,  135135,       1;
%e A200473 1, 126126, 1401400,       1;
%e A200473 1,   6435, 2627625, 2027025,     1;
%p A200473 with(numtheory):
%p A200473 S:= n-> sort([divisors(n)[]]):
%p A200473 T:= (n,k)-> n!/(S(n)[k])!/((n/(S(n)[k]))!)^(S(n)[k]):
%p A200473 seq(seq(T(n, k), k=1..tau(n)), n=1..10);
%t A200473 row[n_] := (n!/#!)/(n/#)!^#& /@ Divisors[n];
%t A200473 Table[row[n], {n, 1, 20}] // Flatten (* _Jean-François Alcover_, Mar 24 2017 *)
%Y A200473 Cf. A200472, A000005 (row lengths).
%Y A200473 Cf. A038041 (row sums).
%K A200473 nonn,tabf
%O A200473 1,7
%A A200473 _Dennis P. Walsh_, Nov 18 2011