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A276922 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.

%I A276922 #15 Sep 08 2021 07:11:13
%S A276922 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 A276922 12,1,0,5040,30240,10290,1540,168,14,1,0,40320,345240,136080,21490,
%U A276922 2464,224,16,1,0,362880,4377240,1977360,326970,38808,3696,288,18,1
%N A276922 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 A276922 Alois P. Heinz, <a href="/A276922/b276922.txt">Rows n = 0..140, flattened</a>
%F A276922 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 A276922 T(n,k) = A276921(n,k) - A276921(n,k-1) for k>0. T(n,0) = A000007(0).
%e A276922 Triangle T(n,k) begins:
%e A276922   1;
%e A276922   0,     1;
%e A276922   0,     2,      1;
%e A276922   0,     6,      6,      1;
%e A276922   0,    24,     42,      8,     1;
%e A276922   0,   120,    330,     80,    10,    1;
%e A276922   0,   720,   2970,    860,   120,   12,   1;
%e A276922   0,  5040,  30240,  10290,  1540,  168,  14,  1;
%e A276922   0, 40320, 345240, 136080, 21490, 2464, 224, 16, 1;
%e A276922   ...
%p A276922 A:= proc(n, k) option remember; `if`(n=0, 1, add(
%p A276922        A(n-i, k)*binomial(n, i), i=1..min(n, k)))
%p A276922     end:
%p A276922 T:= (n, k)-> A(n, k) -`if`(k=0, 0, A(n, k-1)):
%p A276922 seq(seq(T(n, k), k=0..n), n=0..10);
%t A276922 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 A276922 Columns k=0-10 give: A000007, A000142 (for n>0), A320758, A320759, A320760, A320761, A320762, A320763, A320764, A320765, A320766.
%Y A276922 Row sums give A000670.
%Y A276922 T(2n,n) gives A276923.
%Y A276922 Cf. A080510, A276921.
%K A276922 nonn,tabl
%O A276922 0,5
%A A276922 _Alois P. Heinz_, Sep 22 2016