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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-Fran├žois 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<k<=10) give: A007690, A116645, A118807, A184639, A184640, A184641, A184642, A184643, A184644, A184645.

T(2*k+1,k+1) gives A002865.

Sequence in context: A046223 A192181 A073463 * A291958 A127948 A177350

Adjacent sequences:  A183565 A183566 A183567 * A183569 A183570 A183571

KEYWORD

nonn,tabl

AUTHOR

Alois P. Heinz, Jan 05 2011

STATUS

approved

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