OFFSET
0,7
LINKS
Alois P. Heinz, Rows n = 0..200, flattened
Wikipedia, Partition of a set
FORMULA
Sum_{k=1..ceiling(n/2)} k * T(n,k) = A350648(n).
EXAMPLE
T(4,0) = 6: 432|1, 42|31, 42|3|1, 4|31|2, 4|3|21, 4|3|2|1.
T(4,1) = 7: 4321, 43|21, 43|2|1, 421|3, 4|321, 4|32|1, 41|3|2.
T(4,2) = 2: 431|2, 41|32.
T(5,2) = 10: 5431|2, 541|32, 531|42, 51|432, 521|4|3, 5|421|3, 5|42|31, 5|42|3|1, 51|4|32, 51|4|3|2.
T(5,3) = 1: 51|42|3.
Triangle T(n,k) begins:
1;
0, 1;
1, 1;
1, 3, 1;
6, 7, 2;
16, 25, 10, 1;
73, 91, 35, 4;
298, 390, 163, 25, 1;
1453, 1797, 755, 128, 7;
7366, 9069, 3919, 737, 55, 1;
40689, 49106, 21485, 4304, 380, 11;
238258, 284537, 126273, 26695, 2696, 110, 1;
...
MAPLE
b:= proc(n, m) option remember; expand(`if`(n=0, 1, add(
`if`(j=n, x, 1)*b(n-1, max(m, j)), j=1..m+1)))
end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..ceil(n/2)))(b(n, 0)):
seq(T(n), n=0..14);
MATHEMATICA
b[n_, m_] := b[n, m] = Expand[If[n == 0, 1, Sum[
If[j == n, x, 1]*b[n-1, Max[m, j]], {j, 1, m+1}]]];
T[n_] := With[{p = b[n, 0]},
Table[Coefficient[p, x, i], {i, 0, Ceiling[n/2]}]];
Table[T[n], {n, 0, 14}] // Flatten (* Jean-François Alcover, Jan 11 2022, after Alois P. Heinz *)
CROSSREFS
KEYWORD
nonn,tabf
AUTHOR
Alois P. Heinz, Jan 09 2022
STATUS
approved