A059623 As upper right triangle, number of weakly unimodal partitions of n where initial part is k (n >= k >= 1).

1, 1, 1, 2, 1, 1, 4, 2, 1, 1, 8, 3, 2, 1, 1, 15, 5, 3, 2, 1, 1, 27, 8, 5, 3, 2, 1, 1, 47, 13, 7, 5, 3, 2, 1, 1, 79, 21, 11, 7, 5, 3, 2, 1, 1, 130, 33, 16, 11, 7, 5, 3, 2, 1, 1, 209, 52, 24, 15, 11, 7, 5, 3, 2, 1, 1, 330, 80, 35, 22, 15, 11, 7, 5, 3, 2, 1, 1, 512, 122, 52, 31, 22, 15, 11, 7, 5, 3

Offset: 1

Author

Henry Bottomley, Feb 01 2001

Keywords

• nonn
• tabl

Comments

Weakly unimodal means nondecreasing then nonincreasing.

Examples

			Rows are {1,1,2,4,8,15,...}, {1,1,2,3,5,8,...}, {1,1,2,3,5,7,...} etc.
As an upper right triangle:
  1,  1,  2,  4,  8, 15, ...,
      1,  1,  2,  3,  5,  8, ...,
          1,  1,  2,  3,  5,  7, ...,
              ...
As a left downward triangle, it starts:
   1;
   1, 1;
   2, 1, 1;
   4, 2, 1, 1;
   8, 3, 2, 1, 1;
  15, 5, 3, 2, 1, 1;
  27, 8, 5, 3, 2, 1, 1;
  ...
T(9,3)=11 since 9 can be written as 3+6, 3+5+1, 3+4+2, 3+4+1+1, 3+3+3, 3+3+2+1, 3+3+1+1+1, 3+2+2+2, 3+2+2+1+1, 3+2+1+1+1+1 or 3+1+1+1+1+1.
		

Crossrefs

Column sums give A001523. Cf. A008284, A026836, A008284, A059607, A059619.

Formula

T(n, k) = S(n, k) - S(n-k, k) + Sum_j[T(n-k, j)] for j >= k, where S(n, k) = A008284(n, k) = Sum_j[S(n-k, j)] for n>k >= j [note reversal] with S[n, n] = 1.