cp's OEIS Frontend

A262071 Number T(n,k) of ordered partitions of an n-set with nondecreasing block sizes and maximal block size equal to k; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

%I A262071 #20 Oct 26 2018 18:43:27
%S A262071 1,0,1,0,2,1,0,6,3,1,0,24,18,4,1,0,120,90,30,5,1,0,720,630,200,45,6,1,
%T A262071 0,5040,4410,1610,350,63,7,1,0,40320,37800,13440,3290,560,84,8,1,0,
%U A262071 362880,340200,130200,30870,5922,840,108,9,1,0,3628800,3515400,1327200,334950,61992,9870,1200,135,10,1
%N A262071 Number T(n,k) of ordered partitions of an n-set with nondecreasing block sizes and maximal block size equal to k; triangle T(n,k), n>=0, 0<=k<=n, read by rows.
%H A262071 Alois P. Heinz, <a href="/A262071/b262071.txt">Rows n = 0..140, flattened</a>
%F A262071 E.g.f. of column k: x^k * Product_{i=1..k} (i-1)!/(i!-x^i).
%e A262071 T(3,1) = 6: 1|2|3, 1|3|2, 2|1|3, 2|3|1, 3|1|2, 3|2|1.
%e A262071 T(3,2) = 3: 1|23, 2|13, 3|12.
%e A262071 T(3,3) = 1: 123.
%e A262071 Triangle T(n,k) begins:
%e A262071   1;
%e A262071   0,     1;
%e A262071   0,     2,     1;
%e A262071   0,     6,     3,     1;
%e A262071   0,    24,    18,     4,    1;
%e A262071   0,   120,    90,    30,    5,   1;
%e A262071   0,   720,   630,   200,   45,   6,  1;
%e A262071   0,  5040,  4410,  1610,  350,  63,  7, 1;
%e A262071   0, 40320, 37800, 13440, 3290, 560, 84, 8, 1;
%p A262071 b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
%p A262071        b(n, i-1)+`if`(i>n, 0, binomial(n, i)*b(n-i, i))))
%p A262071     end:
%p A262071 T:= (n, k)-> b(n, k) -`if`(k=0, 0, b(n, k-1)):
%p A262071 seq(seq(T(n, k), k=0..n), n=0..12);
%t A262071 b[n_, i_] := b[n, i] = If[n == 0, 1, If[i < 1, 0, b[n, i - 1] + If[i > n, 0, Binomial[n, i]*b[n - i, i]]]]; T[n_, k_] :=  b[n, k] - If[k == 0, 0, b[n, k - 1]]; Table[T[n, k], {n, 0, 12}, {k, 0, n}] // Flatten (* _Jean-François Alcover_, Dec 12 2016, _Alois P. Heinz_ *)
%Y A262071 Columns k=0-10 give: A000007, A000142 (for n>0), A272492, A272493, A272494, A272495, A272496, A272497, A272498, A272499, A272500.
%Y A262071 Main diagonal gives A000012.
%Y A262071 Row sums give A005651.
%Y A262071 T(2n,n) gives A266518.
%Y A262071 Cf. A262072.
%K A262071 nonn,tabl
%O A262071 0,5
%A A262071 _Alois P. Heinz_, Sep 10 2015