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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 (list; table; graph; refs; listen; history; text; internal format)
 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 Columns k=0-10 give: A000041, A183558, A183559, A183560, A183561, A183562, A183563, A183564, A183565, A183566, A183567. Differences between columns 0 and k (0

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Last modified October 16 06:05 EDT 2019. Contains 328046 sequences. (Running on oeis4.)