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Triangular array T(n,k): The differences in the columns of A174551.
0

%I #8 Sep 18 2013 04:41:54

%S 1,0,1,1,1,2,8,7,6,6,81,65,50,36,24,1024,781,570,390,240,120,15625,

%T 11529,8162,5460,3360,1800,720,279936,201811,140070,92526,57120,31920,

%U 15120,5040

%N Triangular array T(n,k): The differences in the columns of A174551.

%C T(n,k) is the number of functions f:{1,2,...,n}->{1,2,...,n} such that the image of f contains {1,2,...,k} but not k+1.

%C Row sums = n^n.

%C Columns are asymptotic to n^n (1/e)(1-(1/e))^k.

%C Sum k*T(n,k) appears to be A055869.

%e Triangle begins:

%e 1;

%e 0, 1;

%e 1, 1, 2;

%e 8, 7, 6, 6;

%e 81,65,50,36,24;

%e 1024,781,570,390,240,120;

%e 15625, 11529, 8162, 5460, 3360, 1800, 720...

%t Table[Append[(-1) Differences[ Table[Sum[(-1)^i Binomial[k, i] (n - i)^n, {i, 0, k}], {k, 0, n}]], n! ], {n, 0, 7}] // Grid

%K nonn,tabl

%O 0,6

%A _Geoffrey Critzer_, Mar 22 2010