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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.
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%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