A076014 Triangle in which m-th entry of n-th row is m^(n-1).

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, 1, 256, 6561, 65536, 390625, 1679616, 5764801, 16777216, 43046721

Offset: 1

Author

Wolfdieter Lang, Oct 02 2002

Keywords

• nonn
• easy
• tabl

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

Examples

			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;
  ...
		

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.

Crossrefs

Cf. A009998, A008279, A008277 (Stirling2).

Cf. A089072.

Programs

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 *)

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.