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%I #38 Dec 13 2022 10:15:45
%S 1,1,2,1,4,9,1,8,27,64,1,16,81,256,625,1,32,243,1024,3125,7776,1,64,
%T 729,4096,15625,46656,117649,1,128,2187,16384,78125,279936,823543,
%U 2097152,1,256,6561,65536,390625,1679616,5764801,16777216,43046721
%N Triangle in which m-th entry of n-th row is m^(n-1).
%C This becomes triangle A009998(n-1, m-1), n >= m >= 1, if the m-th column entries are divided by m^(m-1).
%C Row sums give A076015. The m-th column (without leading zeros) gives (m^(m-1)) powers of m, m >= 1.
%C 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
%C 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
%H Mohammad K. Azarian, <a href="https://doi.org/10.12988/imf.2022.912321">Remarks and Conjectures Regarding Combinatorics of Discrete Partial Functions</a>, Int'l Math. Forum (2022) Vol. 17, No. 3, 129-141. See Theorem 2.5, p. 132.
%F T(n, m) = m^(n-1), n >= m >= 1, otherwise 0.
%F G.f. for m-th column: (m^(m-1))(x^m)/(1-m*x), m >= 1.
%F a(n,m) = Sum_{p=1..m} Stirling2(n,p)*A008279(m-1, p-1), n >= m >= 1, otherwise 0.
%e 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
%e Triangle begins:
%e 1;
%e 1, 2;
%e 1, 4, 9;
%e 1, 8, 27, 64;
%e 1, 16, 81, 256, 625;
%e 1, 32, 243, 1024, 3125, 7776;
%e 1, 64, 729, 4096, 15625, 46656, 117649;
%e 1, 128, 2187, 16384, 78125, 279936, 823543, 2097152;
%e ...
%p seq(seq(m^(n-1),m=1..n),n=1..20); # _Dennis P. Walsh_, Feb 27 2013
%t Table[m^(n-1),{n,10},{m,n}]//Flatten (* _Harvey P. Dale_, May 27 2017 *)
%Y Cf. A009998, A008279, A008277 (Stirling2).
%Y Cf. A089072.
%K nonn,easy,tabl
%O 1,3
%A _Wolfdieter Lang_, Oct 02 2002
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