A132311 Triangle read by rows: T(n,k) is the number of partitions of binomial(n,k) into parts of the first n rows of Pascal's triangle, 0<=k<=n.

0, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 4, 7, 4, 1, 1, 6, 28, 28, 6, 1, 1, 11, 117, 318, 117, 11, 1, 1, 14, 388, 3344, 3344, 388, 14, 1, 1, 21, 1757, 71277, 290521, 71277, 1757, 21, 1, 1, 29, 8270, 2031198, 53679222, 53679222, 2031198, 8270, 29, 1, 1, 42, 40243, 74464383, 19465193506, 147286801214, 19465193506, 74464383, 40243, 42, 1

Offset: 0

Author

Reinhard Zumkeller, Aug 18 2007

Keywords

• nonn
• tabl

Comments

T(n,k) = T(n,n-k).

T(n,0) = 1 for n>0.

A000041(n) - 1 <= T(n,1) <= A000041(n) for n>1.

Examples

			A007318(4,2) = A007318(6,1) = 6: T(4,2) = #{3+3, 3+2+1, 3+1+1+1, 2+2+2, 2+2+1+1, 2+1+1+1+1, 1+1+1+1+1+1} = 7, but T(6,1) = A000041(6) = 11.
Triangle T(n,k) begins:
  0;
  1,  1;
  1,  1,    1;
  1,  2,    2,     1;
  1,  4,    7,     4,      1;
  1,  6,   28,    28,      6,     1;
  1, 11,  117,   318,    117,    11,    1;
  1, 14,  388,  3344,   3344,   388,   14,  1;
  1, 21, 1757, 71277, 290521, 71277, 1757, 21, 1;
  ...
		

Links

• Alois P. Heinz, Rows n = 0..18, flattened
• Index entries for triangles and arrays related to Pascal's triangle

Crossrefs

Cf. A132312, A007318, A126257, A014631.