OFFSET
0,9
COMMENTS
The sequence of column k satisfies a linear recurrence with constant coefficients of order 2^(k-1) for k>0.
LINKS
Alois P. Heinz, Antidiagonals n = 0..45, flattened
Pierpaolo Natalini, Paolo Emilio Ricci, New Bell-Sheffer Polynomial Sets, Axioms 2018, 7(4), 71.
Wikipedia, Partition of a set
FORMULA
A(n,k) = Sum_{j=0..k} A287213(n,j).
EXAMPLE
A(4,0) = 1: 1|2|3|4.
A(4,1) = 8: 1234, 123|4, 12|34, 12|3|4, 1|234, 1|23|4, 1|2|34, 1|2|3|4.
A(4,2) = 13: 1234, 123|4, 124|3, 12|34, 12|3|4, 134|2, 13|24, 13|2|4, 1|234, 1|23|4, 1|24|3, 1|2|34, 1|2|3|4.
Square array A(n,k) begins:
1, 1, 1, 1, 1, 1, 1, 1, ...
1, 1, 1, 1, 1, 1, 1, 1, ...
1, 2, 2, 2, 2, 2, 2, 2, ...
1, 4, 5, 5, 5, 5, 5, 5, ...
1, 8, 13, 15, 15, 15, 15, 15, ...
1, 16, 34, 47, 52, 52, 52, 52, ...
1, 32, 89, 150, 188, 203, 203, 203, ...
1, 64, 233, 481, 696, 825, 877, 877, ...
MAPLE
b:= proc(n, k, l) option remember; `if`(n=0, 1, b(n-1, k, map(x->
`if`(x-n>=k, [][], x), [l[], n]))+add(b(n-1, k, sort(map(x->
`if`(x-n>=k, [][], x), subsop(j=n, l)))), j=1..nops(l)))
end:
A:= (n, k)-> b(n, min(k, n-1), []):
seq(seq(A(n, d-n), n=0..d), d=0..12);
MATHEMATICA
b[0, _, _] = 1; b[n_, k_, l_List] := b[n, k, l] = b[n - 1, k, If[# - n >= k, Nothing, #]& /@ Append[l, n]] + Sum[b[n - 1, k, Sort[If[# - n >= k, Nothing, #]& /@ ReplacePart[l, j -> n]]], {j, 1, Length[l]}];
A[n_, k_] := b[n, Min[k, n - 1], {}];
Table[A[n, d - n], {d, 0, 12}, {n, 0, d}] // Flatten (* Jean-François Alcover, Apr 30 2018, after Alois P. Heinz *)
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Alois P. Heinz, May 21 2017
STATUS
approved