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A285824 Number T(n,k) of ordered set partitions of [n] into k blocks such that equal-sized blocks are ordered with increasing least elements; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

%I A285824 #21 Feb 17 2023 11:08:42
%S A285824 1,0,1,0,1,1,0,1,6,1,0,1,11,18,1,0,1,30,75,40,1,0,1,52,420,350,75,1,0,
%T A285824 1,126,1218,3080,1225,126,1,0,1,219,4242,17129,15750,3486,196,1,0,1,
%U A285824 510,14563,82488,152355,63756,8526,288,1,0,1,896,42930,464650,1049895,954387,217560,18600,405,1
%N A285824 Number T(n,k) of ordered set partitions of [n] into k blocks such that equal-sized blocks are ordered with increasing least elements; triangle T(n,k), n>=0, 0<=k<=n, read by rows.
%H A285824 Alois P. Heinz, <a href="/A285824/b285824.txt">Rows n = 0..140, flattened</a>
%H A285824 Wikipedia, <a href="https://en.wikipedia.org/wiki/Partition_of_a_set">Partition of a set</a>
%e A285824 T(3,1) = 1: 123.
%e A285824 T(3,2) = 6: 1|23, 23|1, 2|13, 13|2, 3|12, 12|3.
%e A285824 T(3,3) = 1: 1|2|3.
%e A285824 Triangle T(n,k) begins:
%e A285824   1;
%e A285824   0, 1;
%e A285824   0, 1,   1;
%e A285824   0, 1,   6,    1;
%e A285824   0, 1,  11,   18,     1;
%e A285824   0, 1,  30,   75,    40,     1;
%e A285824   0, 1,  52,  420,   350,    75,    1;
%e A285824   0, 1, 126, 1218,  3080,  1225,  126,   1;
%e A285824   0, 1, 219, 4242, 17129, 15750, 3486, 196, 1;
%e A285824   ...
%p A285824 b:= proc(n, i, p) option remember; expand(`if`(n=0 or i=1,
%p A285824       (p+n)!/n!*x^n, add(b(n-i*j, i-1, p+j)*x^j*combinat
%p A285824       [multinomial](n, n-i*j, i$j)/j!^2, j=0..n/i)))
%p A285824     end:
%p A285824 T:= n-> (p-> seq(coeff(p, x, i), i=0..n))(b(n$2, 0)):
%p A285824 seq(T(n), n=0..12);
%t A285824 multinomial[n_, k_List] := n!/Times @@ (k!);
%t A285824 b[n_, i_, p_] := b[n, i, p] = Expand[If[n == 0 || i == 1, (p + n)!/n!*x^n, Sum[b[n-i*j, i-1, p+j]*x^j*multinomial[n, Join[{n-i*j}, Table[i, j]]]/ j!^2, {j, 0, n/i}]]];
%t A285824 T[n_] := Function[p, Table[Coefficient[p, x, i], {i, 0, n}]][b[n, n, 0]];
%t A285824 Table[T[n], {n, 0, 12}] // Flatten (* _Jean-François Alcover_, Apr 28 2018, after _Alois P. Heinz_ *)
%Y A285824 Columns k=0-10 give: A000007, A057427, A285917, A285918, A285919, A285920, A285921, A285922, A285923, A285924, A285925.
%Y A285824 Main diagonal and first lower diagonal give: A000012, A002411.
%Y A285824 Row sums give A120774.
%Y A285824 T(2n,n) gives A285926.
%Y A285824 Cf. A048993, A131689, A226874, A285849.
%K A285824 nonn,tabl
%O A285824 0,9
%A A285824 _Alois P. Heinz_, Apr 27 2017