OFFSET
0,4
COMMENTS
The blocks are ordered with increasing least elements.
LINKS
Alois P. Heinz, Rows n = 0..27, flattened
Wikipedia, Partition of a set
FORMULA
Sum_{k=0..max(0,n-1)} k * T(n,k) = A363549(n).
EXAMPLE
T(4,1) = 3: 134|2, 13|24, 13|2|4.
T(4,2) = 4: 124|3, 14|23, 14|2|3, 1|24|3.
T(4,3) = 8: 1234, 123|4, 12|34, 12|3|4, 1|234, 1|23|4, 1|2|34, 1|2|3|4.
T(5,2) = 18: 1245|3, 124|35, 124|3|5, 134|25, 134|2|5, 13|245, 13|24|5, 13|2|45, 13|2|4|5, 14|235, 14|23|5, 14|25|3, 14|2|35, 14|2|3|5, 15|24|3, 1|245|3, 1|24|35, 1|24|3|5.
T(5,4) = 18: 12345, 1234|5, 123|45, 123|4|5, 12|345, 12|34|5, 12|3|45, 12|3|4|5, 145|23, 1|2345, 1|234|5, 1|23|45, 1|23|4|5, 145|2|3, 1|2|345, 1|2|34|5, 1|2|3|45, 1|2|3|4|5.
Triangle T(n,k) begins:
1;
1;
0, 2;
0, 1, 4;
0, 3, 4, 8;
0, 2, 18, 14, 18;
0, 7, 27, 87, 42, 40;
0, 5, 102, 162, 360, 147, 101;
0, 20, 179, 866, 931, 1456, 434, 254;
0, 15, 675, 1746, 5836, 4755, 5778, 1619, 723;
0, 67, 1321, 9087, 16416, 36031, 22893, 23052, 5044, 2064;
...
MAPLE
b:= proc(l, i, t) option remember; expand(`if`(l=[], 1,
add((f-> b(subsop(j=[][], l), j, `if`(f, 1-t, t))*
`if`(f, x, 1))(l[j]=t), j=[1, $i..nops(l)])))
end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(
b([ seq(irem(i, 2), i=2..n)], 1, 0)):
seq(T(n), n=0..12);
MATHEMATICA
b[l_, i_, t_] := b[l, i, t] = Expand[If[l == {}, 1, Sum[Function[f, b[ReplacePart[l, j -> Nothing], j, If[f, 1 - t, t]]*If[f, x, 1]][l[[j]] == t], {j, Join[{1}, Range[i, Length@l]]}]]];
T[n_] := CoefficientList[b[ Table[Mod[i, 2], {i, 2, n}], 1, 0], x];
Table[T[n], {n, 0, 12}] // Flatten (* Jean-François Alcover, Nov 17 2023, after Alois P. Heinz *)
CROSSREFS
KEYWORD
nonn,tabf
AUTHOR
Alois P. Heinz, Jun 07 2023
STATUS
approved