A059607 As an upper right triangle, number of distinct partitions of n where the highest part is k (0<=k<=n).

1, 0, 1, 0, 0, 1, 0, 0, 1, 1, 0, 0, 0, 1, 1, 0, 0, 0, 1, 1, 1, 0, 0, 0, 1, 1, 1, 1, 0, 0, 0, 0, 2, 1, 1, 1, 0, 0, 0, 0, 1, 2, 1, 1, 1, 0, 0, 0, 0, 1, 2, 2, 1, 1, 1, 0, 0, 0, 0, 1, 2, 2, 2, 1, 1, 1, 0, 0, 0, 0, 0, 2, 3, 2, 2, 1, 1, 1, 0, 0, 0, 0, 0, 2, 3, 3, 2, 2, 1, 1, 1, 0, 0, 0, 0, 0, 1, 3, 4, 3, 2, 2, 1, 1, 1

Offset

0,33

Links

Table of n, a(n) for n=0..104.

Formula

T(n, k) =sum_j[T(n-k, j)] for k>j with T(0, 0)=1

Example

Rows are {1,0,0,0,...}, {1,0,0,0,...}, {1,1,0,0,...}, {1,1,1,1,...}, {1,1,1,2,...} etc. T(7,4)=2 since 7 can be written as 4+3 or 4+2+1. T(12,6)=3 since 12 can be written as 6+5+1 or 6+4+2 or 6+3+2+1.

Mathematica

t[n_?Positive, k_] := t[n, k] = Sum[t[n-k, j], {j, 0, k-1}]; t[0, 0] = 1; t[0, _] = 0; t[_?Negative, _] = 0; Table[ t[n, k], {n, 0, 13}, {k, 0, n}] // Flatten (* Jean-François Alcover, Sep 11 2012 *)

Crossrefs

As upper right triangle, row sum is A011782, column sum is A000009, column maximum is A025591 (offset), row maximum is A026839 (offset). Cf. A026836 for this triangle starting at (1, 1) rather than (0, 0).

Sequence in context: A086010 A321929 A089198 * A176724 A015318 A026836

Adjacent sequences: A059604 A059605 A059606 * A059608 A059609 A059610

Keyword

nonn,tabl

Author

Henry Bottomley, Jan 30 2001

Status

approved