OFFSET
0,4
COMMENTS
All parts of a number partition with the same value form a clique. The size of a clique is the number of elements in the clique. Each partition has a clique of size 0.
LINKS
Alois P. Heinz, Rows n = 0..140, flattened
FORMULA
G.f. of column k: (1-Product_{j>0} (1-x^(k*j)+x^((k+1)*j))) / (Product_{j>0} (1-x^j)).
EXAMPLE
T(5,2) = 2, because 2 (of 7) partitions of 5 contain (at least) one clique of size 2: [1,2,2], [1,1,3].
Triangle T(n,k) begins:
1;
1, 1;
2, 1, 1;
3, 2, 0, 1;
5, 3, 2, 0, 1;
7, 6, 2, 1, 0, 1;
11, 7, 3, 2, 1, 0, 1;
15, 13, 5, 3, 1, 1, 0, 1;
MAPLE
b:= proc(n, i, k) option remember; `if`(n=0, [1, 0], `if`(i<1, [0, 0],
add((l->`if`(j=k, [l[1]$2], l))(b(n-i*j, i-1, k)), j=0..n/i)))
end:
T:= (n, k)-> (l-> l[`if`(k=0, 1, 2)])(b(n, n, k)):
seq(seq(T(n, k), k=0..n), n=0..12);
MATHEMATICA
b[n_, i_, k_] := b[n, i, k] = If[n == 0, {1, 0}, If[i < 1, {0, 0}, Sum[Function[l, If[j == k, {l[[1]], l[[1]]}, l]][b[n - i*j, i-1, k]], {j, 0, n/i}]] ]; t[n_, k_] := Function[l, l[[If[k == 0, 1, 2]]]][b[n, n, k]]; Table[Table[t[n, k], {k, 0, n}], {n, 0, 12}] // Flatten (* Jean-François Alcover, Dec 16 2013, translated from Maple *)
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Alois P. Heinz, Jan 05 2011
STATUS
approved