A194710 Triangle read by rows: T(k,m) = number of occurrences of k in the last section of the set of partitions of (10 + m).

42, 15, 27, 10, 14, 18, 5, 10, 10, 17, 4, 5, 8, 10, 15, 2, 5, 4, 8, 9, 14, 2, 2, 4, 5, 7, 9, 13, 1, 2, 2, 4, 4, 8, 8, 13, 1, 1, 2, 2, 4, 4, 7, 9, 12, 0, 1, 1, 2, 2, 4, 4, 7, 8, 13, 1, 0, 1, 1, 2, 2, 4, 4, 7, 8, 12, 0, 1, 0, 1, 1, 2, 2, 4, 4, 7, 8, 12

Offset: 1

Author

Omar E. Pol, Feb 05 2012

Keywords

• nonn
• tabl

Comments

Sub-triangle of A182703 and also of A194812. Note that the sum of row k is also the number of partitions of 10. For further information see A182703 and A135010.

Examples

			Triangle begins:
  42;
  15, 27;
  10, 14, 18;
   5, 10, 10, 17;
   4,  5,  8, 10, 15;
   2,  5,  4,  8,  9, 14;
   2,  2,  4,  5,  7,  9, 13;
   1,  2,  2,  4,  4,  8,  8, 13;
   1,  1,  2,  2,  4,  4,  7,  9, 12;
   0,  1,  1,  2,  2,  4,  4,  7,  8, 13;
   1,  0,  1,  1,  2,  2,  4,  4,  7,  8, 12;
   0,  1,  0,  1,  1,  2,  2,  4,  4,  7,  8, 12;
   0,  0,  1,  0,  1,  1,  2,  2,  4,  4,  7,  8, 12;
   0,  0,  0,  1,  0,  1,  1,  2,  2,  4,  4,  7,  8, 12;
  ...
For k = 1 and m = 1; T(1,1) = 42 because there are 42 parts of size 1 in the last section of the set of partitions of 11, since 10 + m = 11, so a(1) = 42. For k = 2 and m = 1; T(2,1) = 15 because there are 15 parts of size 2 in the last section of the set of partitions of 11, since 10 + m = 11, so a(2) = 15.
		

Crossrefs

Always the sum of row k = p(10) = A000041(10) = 42.

The first (0-10) members of this family of triangles are A023531, A129186, A194702-A194709, this sequence.

Cf. A002865, A135010, A138121, A194812.

Formula

T(k,m) = A182703(10+m,k), with T(k,m) = 0 if k > 10+m.

T(k,m) = A194812(10+m,k).

Beginning with row k=11 each row starts with (k-11) 0's and ends with the subsequence 1, 0, 1, 1, 2, 2, 4, 4, 7, 8, 12, the initial terms of A002865. - Alois P. Heinz, Feb 15 2012