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A183568
Triangle T(n,k), n>=0, 0<=k<=n, read by rows: T(n,k) is the number of partitions of n containing a clique of size k.
22
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, 22, 16, 9, 3, 3, 1, 1, 0, 1, 30, 25, 10, 6, 3, 2, 1, 1, 0, 1, 42, 33, 16, 8, 5, 3, 2, 1, 1, 0, 1, 56, 49, 23, 13, 6, 5, 2, 2, 1, 1, 0, 1, 77, 61, 31, 15, 10, 5, 5, 2, 2, 1, 1, 0, 1
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
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
Differences between columns 0 and k (0<k<=10) give: A007690, A116645, A118807, A184639, A184640, A184641, A184642, A184643, A184644, A184645.
T(2*k+1,k+1) gives A002865.
Sequence in context: A338398 A192181 A073463 * A291958 A127948 A177350
KEYWORD
nonn,tabl
AUTHOR
Alois P. Heinz, Jan 05 2011
STATUS
approved