A299779 Triangle read by rows: T(n,k) is the total number of cliques of size k in all partitions of all positive integers <= n.

1, 2, 1, 5, 1, 1, 9, 3, 1, 1, 17, 5, 2, 1, 1, 28, 9, 4, 2, 1, 1, 47, 14, 7, 3, 2, 1, 1, 73, 24, 10, 6, 3, 2, 1, 1, 114, 35, 17, 9, 5, 3, 2, 1, 1, 170, 55, 25, 14, 8, 5, 3, 2, 1, 1, 253, 80, 38, 20, 13, 7, 5, 3, 2, 1, 1, 365, 118, 55, 31, 18, 12, 7, 5, 3, 2, 1, 1, 525, 167, 80, 44, 27, 17, 11, 7, 5, 3, 2, 1, 1

Offset: 1

Author

Omar E. Pol, Apr 04 2018

Keywords

• nonn
• tabl

Comments

Column k gives the partial sums of the k-th column of triangle A197126.

Examples

			Triangle begins:
    1;
    2,   1;
    5,   1,  1;
    9,   3,  1,  1;
   17,   5,  2,  1,  1;
   28,   9,  4,  2,  1,  1;
   47,  14,  7,  3,  2,  1,  1;
   73,  24, 10,  6,  3,  2,  1,  1;
  114,  35, 17,  9,  5,  3,  2,  1,  1;
  170,  55, 25, 14,  8,  5,  3,  2,  1,  1;
  253,  80, 38, 20, 13,  7,  5,  3,  2,  1,  1;
  365, 118, 55, 31, 18, 12,  7,  5,  3,  2,  1,  1;
  525, 167, 80, 44, 27, 17, 11,  7,  5,  3,  2,  1,  1;
...
		

Links

• Alois P. Heinz, Rows n = 1..150, flattened

Crossrefs

Column 1 gives A000097.

Row sums give A014153.

Cf. A000041, A000070, A197126, A284870.

Programs

Maple

b:= proc(n, p, k) option remember; `if`(n=0, [1, 0], `if`(p<1, [0, 0],
      add((l->`if`(m=k, l+[0, l[1]], l))(b(n-p*m, p-1, k)), m=0..n/p)))
    end:
T:= proc(n, k) option remember;
      b(n$2, k)[2]+`if`(nAlois P. Heinz, Apr 27 2018

Mathematica

b[n_, p_, k_] := b[n, p, k] = If[n == 0, {1, 0}, If[p < 1, {0, 0}, Sum[ Function[l, If[m==k, l+{0, l[[1]]}, l]][b[n-p*m, p-1, k]], {m, 0, n/p}]]];
T[n_, k_] := b[n, n, k][[2]] + If[n < k, 0, T[n-1, k]];
Table[Table[T[n, k], {k, 1, n}], {n, 1, 20}] // Flatten (* Jean-François Alcover, Dec 07 2019, after Alois P. Heinz *)

Formula

T(n,k) = Sum_{j=k..n} A197126(j,k).

T(2n+1,n+1) = A000041(n). - Alois P. Heinz, Apr 27 2018

Sum_{k=1..n} k * T(n,k) = A284870(n). - Alois P. Heinz, May 14 2018