A089246 Triangular array read by rows: a(n, k) is the number of ordered m-tuples of positive integers (x_1, ..., x_m) such that max x_i = n+1-m and there are k ones (0 <= k <= n).
1, 0, 1, 1, 0, 1, 2, 2, 0, 1, 5, 5, 3, 0, 1, 14, 15, 9, 4, 0, 1, 43, 50, 31, 14, 5, 0, 1, 144, 180, 118, 54, 20, 6, 0, 1, 523, 695, 481, 229, 85, 27, 7, 0, 1, 2048, 2869, 2081, 1035, 395, 125, 35, 8, 0, 1, 8597, 12616, 9535, 4929, 1951, 629, 175, 44, 9, 0, 1, 38486, 58862
Offset: 0
Examples
a(5, 0) = 14: (5), (4,2), (2,4), (4,3), (3,4), (4,4),
(3,2,2), (2,3,2), (2,2,3), (3,3,2), (3,2,3), (2,3,3), (3,3,3), (2,2,2,2).
a(5, 1) = 15: (4,1), (1,4), (3,3,1), (3,1,3), (1,3,3), 6 permutations of (3,2,1) and 4 permutations of (2,2,2,1).
Triangle starts:
[0] 1
[1] 0, 1
[2] 1, 0, 1
[3] 2, 2, 0, 1
[4] 5, 5, 3, 0, 1
[5] 14, 15, 9, 4, 0, 1
[6] 43, 50, 31, 14, 5, 0, 1
[7] 144, 180, 118, 54, 20, 6, 0, 1
[8] 523, 695, 481, 229, 85, 27, 7, 0, 1
[9] 2048, 2869, 2081, 1035, 395, 125, 35, 8, 0, 1
Links
- Mathew Englander, Comments on A101494 and A089246, and related sequences
Formula
From Mathew Englander, Feb 25 2021: (Start)
T(n,k) = 0^(n-k) + Sum_{m = k..n-1} C(m,k) * ((n-m)^(m-k) - (n-1-m)^(m-k)).
T(n,k) = Sum_{j = k+1..n-1} C(j,k)*Sum_{i = j..n-1} T(i,j) for 0 <= k < n-1; T(k+1,k)=0 and T(k,k)=1 for k>=0.
G.f. of row n: 1 + Sum_{i = 1..n} (x+n-i)^(i-1)*(x+n-i-1). (End)
Extensions
Edited and extended by David Wasserman, Sep 07 2005
Comments