%I #5 Jan 18 2015 21:11:59
%S 1,1,-1,1,0,1,1,1,-1,-1,1,2,-1,2,1,1,3,1,-1,-3,-1,1,4,5,-4,5,4,1,1,5,
%T 11,-1,1,-11,-5,-1,1,6,19,14,-15,14,19,6,1,1,7,29,47,-19,19,-47,-29,
%U -7,-1,1,8,41,104,37,-56,37,104,41,8,1
%N Square array read by ascending antidiagonals, T(n, k) = k!*[x^k](exp(-x) *sum(j=0..n, C(n,j)*x^j)), n>=0, k>=0.
%F T(n,n) = A009940(n).
%e Square array starts:
%e [n\k][0 1 2 3 4 5 6]
%e [0] 1, -1, 1, -1, 1, -1, 1, ...
%e [1] 1, 0, -1, 2, -3, 4, -5, ...
%e [2] 1, 1, -1, -1, 5, -11, 19, ...
%e [3] 1, 2, 1, -4, 1, 14, -47, ...
%e [4] 1, 3, 5, -1, -15, 19, 37, ...
%e [5] 1, 4, 11, 14, -19, -56, 151, ...
%e [6] 1, 5, 19, 47, 37, -151, -185, ...
%e The first few rows as a triangle:
%e 1,
%e 1, -1,
%e 1, 0, 1,
%e 1, 1, -1, -1,
%e 1, 2, -1, 2, 1,
%e 1, 3, 1, -1, -3, -1,
%e 1, 4, 5, -4, 5, 4, 1.
%p T := (n,k) -> k!*coeff(series(exp(-x)*add(binomial(n,j)*x^j, j=0..n), x, k+1), x, k): for n from 0 to 6 do lprint(seq(T(n,k),k=0..6)) od;
%Y Cf. A009940.
%K sign,tabl
%O 0,12
%A _Peter Luschny_, Jan 18 2015