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A276921 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.

%I A276921 #20 Jan 01 2022 04:44:52
%S A276921 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 A276921 13,74,450,720,0,1,1,3,13,75,530,3690,5040,0,1,1,3,13,75,540,4550,
%U A276921 35280,40320,0,1,1,3,13,75,541,4670,45570,385560,362880,0
%N A276921 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 A276921 Alois P. Heinz, <a href="/A276921/b276921.txt">Antidiagonals n = 0..140, flattened</a>
%H A276921 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 A276921 E.g.f. of column k: 1/(1-Sum_{i=1..k} x^i/i!).
%F A276921 A(n,k) = Sum_{j=0..k} A276922(n,j).
%e A276921 Square array A(n,k) begins:
%e A276921   1,    1,     1,     1,     1,     1,     1,     1, ...
%e A276921   0,    1,     1,     1,     1,     1,     1,     1, ...
%e A276921   0,    2,     3,     3,     3,     3,     3,     3, ...
%e A276921   0,    6,    12,    13,    13,    13,    13,    13, ...
%e A276921   0,   24,    66,    74,    75,    75,    75,    75, ...
%e A276921   0,  120,   450,   530,   540,   541,   541,   541, ...
%e A276921   0,  720,  3690,  4550,  4670,  4682,  4683,  4683, ...
%e A276921   0, 5040, 35280, 45570, 47110, 47278, 47292, 47293, ...
%p A276921 A:= proc(n, k) option remember; `if`(n=0, 1, add(
%p A276921        A(n-i, k)*binomial(n, i), i=1..min(n, k)))
%p A276921     end:
%p A276921 seq(seq(A(n, d-n), n=0..d), d=0..12);
%t A276921 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 A276921 Columns k=0..10 give: A000007, A000142, A080599, A189886, A276924, A276925, A276926, A276927, A276928, A276929, A276930.
%Y A276921 Main diagonal gives A000670.
%Y A276921 Cf. A276922.
%K A276921 nonn,tabl
%O A276921 0,9
%A A276921 _Alois P. Heinz_, Sep 22 2016