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%I #15 Mar 08 2018 12:50:59
%S 2,2,1,2,0,0,2,-1,-1,3,2,-2,0,2,18,2,-3,3,1,7,95,2,-4,8,-6,2,34,592,2,
%T -5,15,-25,15,13,218,4277,2,-6,24,-62,82,-28,80,1574,35010,2,-7,35,
%U -123,263,-269,106,579,12879,320589
%N Rectangular array read by antidiagonals: a(m,n) = 2 * Integral_{t>=0} T_n((t-m)/2)*exp(-t)*dt, m>=0, n>=0, where T_n(x) is n-th Chebyshev polynomial of first kind.
%C Although negative values of m are not present here or in A300480, the two arrays are connected with the formula: a(m,n) = A300480(-m,n). Thus, they essentially represent two "halves" of the same array indexed by integers m.
%C a(m,n) is a polynomial in m of degree n.
%C For any integers m>=0, n>=0, 2 * Integral_{t=-m..m} T_n(t/2)*exp(-t)*dt = 4 * Integral_{z=-m/2..m/2} T_n(z)*exp(-2*z)*dz = a(m,n)*exp(m) - A300480(m,n)*exp(-m).
%F a(m,n) = A300480(-m,n) = Sum_{i=0..n} A127672(n,i) * i! * Sum_{j=0..i} (-m)^j/j!.
%F a(m,n) = Sum_{i=0..n} A127672(n,i) * A292977(i,m).
%e Array starts with:
%e m=0: 2, 1, 0, 3, 18, 95, 592, ...
%e m=1: 2, 0, -1, 2, 7, 34, 218, ...
%e m=2: 2, -1, 0, 1, 2, 13, 80, ...
%e m=3: 2, -2, 3, -6, 15, -28, 106, ...
%e m=4: 2, -3, 8, -25, 82, -269, 920, ...
%e ...
%o (PARI) { A300481(m,n) = A300480(-m,n); } \\ see code in A300480
%Y Values for m<=0 are given in A300480.
%Y Rows: A300482 (m=0), A300485 (m=1), A102761 (m=2), A300483 (m=-1), A300484 (m=-2).
%Y Columns (up to signs and offset): A007395 (n=0), A000027 (n=1), A005563 (n=2).
%Y Cf. A000179 (almost row m=2), A127672, A156995.
%K sign,tabl
%O 0,1
%A _Max Alekseyev_, Mar 06 2018
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