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A174552 Triangular array T(n,k): The differences in the columns of A174551.

%I A174552 #8 Sep 18 2013 04:41:54
%S A174552 1,0,1,1,1,2,8,7,6,6,81,65,50,36,24,1024,781,570,390,240,120,15625,
%T A174552 11529,8162,5460,3360,1800,720,279936,201811,140070,92526,57120,31920,
%U A174552 15120,5040
%N A174552 Triangular array T(n,k): The differences in the columns of A174551.
%C A174552 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 A174552 Row sums = n^n.
%C A174552 Columns are asymptotic to n^n (1/e)(1-(1/e))^k.
%C A174552 Sum k*T(n,k) appears to be A055869.
%e A174552 Triangle begins:
%e A174552 1;
%e A174552 0, 1;
%e A174552 1, 1, 2;
%e A174552 8, 7, 6, 6;
%e A174552 81,65,50,36,24;
%e A174552 1024,781,570,390,240,120;
%e A174552 15625, 11529, 8162, 5460, 3360, 1800, 720...
%t A174552 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 A174552 nonn,tabl
%O A174552 0,6
%A A174552 _Geoffrey Critzer_, Mar 22 2010