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Triangular array read by rows, T(n,k) is the number of functions from {1,2,...,n} into {1,2,...,n} with maximum value of k.
7

%I #19 Sep 08 2022 08:46:00

%S 1,1,3,1,7,19,1,15,65,175,1,31,211,781,2101,1,63,665,3367,11529,31031,

%T 1,127,2059,14197,61741,201811,543607,1,255,6305,58975,325089,1288991,

%U 4085185,11012415

%N Triangular array read by rows, T(n,k) is the number of functions from {1,2,...,n} into {1,2,...,n} with maximum value of k.

%C Row sums = A000312.

%C Main diagonal = A045531.

%H Vincenzo Librandi, <a href="/A199656/b199656.txt">Rows n = 1..60, flattened</a>

%F T(n,k) = k^n-(k-1)^n.

%e Triangle begins:

%e 1

%e 1 3

%e 1 7 19

%e 1 15 65 175

%e 1 31 211 781 2101

%e 1 63 665 3367 11529 31031

%e 1 127 2059 14197 61741 201811 543607 etc.

%t Table[Table[(1-((i-1)/i)^n) i^n,{i,1,n}],{n,1,8}]//Grid

%t Flatten[Table[k^n - (k-1)^n, {n, 0, 10}, {k, 1, n}]] (* _Vincenzo Librandi_, Jan 28 2013 *)

%o (Magma) /* As triangle: */ [[k^n - (k-1)^n: k in [1..n]]: n in [1..9]]; // _Vincenzo Librandi_, Jan 28 2013

%Y Cf. A022522, A022523.

%K nonn,easy,tabl

%O 1,3

%A _Geoffrey Critzer_, Nov 08 2011