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A059607
As an upper right triangle, number of distinct partitions of n where the highest part is k (0<=k<=n).
4
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
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
KEYWORD
nonn,tabl
AUTHOR
Henry Bottomley, Jan 30 2001
STATUS
approved