OFFSET
1,3
COMMENTS
This becomes triangle A009998(n-1, m-1), n >= m >= 1, if the m-th column entries are divided by m^(m-1).
Row sums give A076015. The m-th column (without leading zeros) gives (m^(m-1)) powers of m, m >= 1.
T(n,m) is the number of functions f:[n-1]->[(n-1)m] such that f(x)=k*x for some positive integer k <= m. Since there exactly m choices for each of the (n-1) images under f, we obtain T(n,m) = m^(n-1). - Dennis P. Walsh, Feb 27 2013
T(n+1,m+1) = (m+1)^n is the number of partial functions from an n-element set to an m-element set, n >= m >= 0. - Mohammad K. Azarian, Jun 28 2021
LINKS
Mohammad K. Azarian, Remarks and Conjectures Regarding Combinatorics of Discrete Partial Functions, Int'l Math. Forum (2022) Vol. 17, No. 3, 129-141. See Theorem 2.5, p. 132.
FORMULA
T(n, m) = m^(n-1), n >= m >= 1, otherwise 0.
G.f. for m-th column: (m^(m-1))(x^m)/(1-m*x), m >= 1.
a(n,m) = Sum_{p=1..m} Stirling2(n,p)*A008279(m-1, p-1), n >= m >= 1, otherwise 0.
EXAMPLE
For example, T(3,2)=4 since there are exactly 4 functions f from {1,2} to {1,2,3,4} that satisfy f(x)=x or f(x)=2x. If we specify each function by the ordered pair (f(1),f(2)), the four functions are (1,2), (1,4), (2,2), and (2,4). - Dennis P. Walsh, Feb 27 2013
Triangle begins:
1;
1, 2;
1, 4, 9;
1, 8, 27, 64;
1, 16, 81, 256, 625;
1, 32, 243, 1024, 3125, 7776;
1, 64, 729, 4096, 15625, 46656, 117649;
1, 128, 2187, 16384, 78125, 279936, 823543, 2097152;
...
MAPLE
seq(seq(m^(n-1), m=1..n), n=1..20); # Dennis P. Walsh, Feb 27 2013
MATHEMATICA
Table[m^(n-1), {n, 10}, {m, n}]//Flatten (* Harvey P. Dale, May 27 2017 *)
CROSSREFS
KEYWORD
AUTHOR
Wolfdieter Lang, Oct 02 2002
STATUS
approved