A276922 - OEIS

%I #15 Sep 08 2021 07:11:13

%S 1,0,1,0,2,1,0,6,6,1,0,24,42,8,1,0,120,330,80,10,1,0,720,2970,860,120,

%T 12,1,0,5040,30240,10290,1540,168,14,1,0,40320,345240,136080,21490,

%U 2464,224,16,1,0,362880,4377240,1977360,326970,38808,3696,288,18,1

%N Number T(n,k) of ordered set partitions of [n] where the maximal block size equals k; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

%H Alois P. Heinz, <a href="/A276922/b276922.txt">Rows n = 0..140, flattened</a>

%F E.g.f. for column k>0: 1/(1-Sum_{i=1..k} x^i/i!) - 1/(1-Sum_{i=1..k-1} x^i/i!).

%F T(n,k) = A276921(n,k) - A276921(n,k-1) for k>0. T(n,0) = A000007(0).

%e Triangle T(n,k) begins:

%e   1;

%e   0,     1;

%e   0,     2,      1;

%e   0,     6,      6,      1;

%e   0,    24,     42,      8,     1;

%e   0,   120,    330,     80,    10,    1;

%e   0,   720,   2970,    860,   120,   12,   1;

%e   0,  5040,  30240,  10290,  1540,  168,  14,  1;

%e   0, 40320, 345240, 136080, 21490, 2464, 224, 16, 1;

%e   ...

%p A:= proc(n, k) option remember; `if`(n=0, 1, add(

%p        A(n-i, k)*binomial(n, i), i=1..min(n, k)))

%p     end:

%p T:= (n, k)-> A(n, k) -`if`(k=0, 0, A(n, k-1)):

%p seq(seq(T(n, k), k=0..n), n=0..10);

%t A[n_, k_] := A[n, k] = If[n == 0, 1, Sum[A[n - i, k]*Binomial[n, i], {i, 1, Min[n, k]}]]; T[n_, k_] :=  A[n, k] - If[k == 0, 0, A[n, k - 1]]; Table[T[n, k], {n, 0, 10}, { k, 0, n}] // Flatten (* _Jean-François Alcover_, Feb 11 2017, translated from Maple *)

%Y Columns k=0-10 give: A000007, A000142 (for n>0), A320758, A320759, A320760, A320761, A320762, A320763, A320764, A320765, A320766.

%Y Row sums give A000670.

%Y T(2n,n) gives A276923.

%Y Cf. A080510, A276921.

%K nonn,tabl

%O 0,5

%A _Alois P. Heinz_, Sep 22 2016