A276921 - OEIS

%I #20 Jan 01 2022 04:44:52

%S 1,1,0,1,1,0,1,1,2,0,1,1,3,6,0,1,1,3,12,24,0,1,1,3,13,66,120,0,1,1,3,

%T 13,74,450,720,0,1,1,3,13,75,530,3690,5040,0,1,1,3,13,75,540,4550,

%U 35280,40320,0,1,1,3,13,75,541,4670,45570,385560,362880,0

%N Number A(n,k) of ordered set partitions of [n] with at most k elements per block; square array A(n,k), n>=0, k>=0, read by antidiagonals.

%H Alois P. Heinz, <a href="/A276921/b276921.txt">Antidiagonals n = 0..140, flattened</a>

%H Daniel Birmajer, Juan B. Gil, David S. Kenepp, and Michael D. Weiner, <a href="https://arxiv.org/abs/2108.04302">Restricted generating trees for weak orderings</a>, arXiv:2108.04302 [math.CO], 2021.

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

%F A(n,k) = Sum_{j=0..k} A276922(n,j).

%e Square array A(n,k) begins:

%e   1,    1,     1,     1,     1,     1,     1,     1, ...

%e   0,    1,     1,     1,     1,     1,     1,     1, ...

%e   0,    2,     3,     3,     3,     3,     3,     3, ...

%e   0,    6,    12,    13,    13,    13,    13,    13, ...

%e   0,   24,    66,    74,    75,    75,    75,    75, ...

%e   0,  120,   450,   530,   540,   541,   541,   541, ...

%e   0,  720,  3690,  4550,  4670,  4682,  4683,  4683, ...

%e   0, 5040, 35280, 45570, 47110, 47278, 47292, 47293, ...

%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 seq(seq(A(n, d-n), n=0..d), d=0..12);

%t A[n_, k_] := A[n, k] = If[n==0, 1, Sum[A[n-i, k]*Binomial[n, i], {i, 1, Min[n, k]}]]; Table[A[n, d-n], {d, 0, 12}, {n, 0, d}] // Flatten (* _Jean-François Alcover_, Feb 03 2017, translated from Maple *)

%Y Columns k=0..10 give: A000007, A000142, A080599, A189886, A276924, A276925, A276926, A276927, A276928, A276929, A276930.

%Y Main diagonal gives A000670.

%Y Cf. A276922.

%K nonn,tabl

%O 0,9

%A _Alois P. Heinz_, Sep 22 2016