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%I #20 Mar 08 2018 13:20:01
%S 2,2,1,2,2,0,2,3,3,3,2,4,8,10,18,2,5,15,29,47,95,2,6,24,66,130,256,
%T 592,2,7,35,127,327,697,1610,4277,2,8,48,218,722,1838,4376,11628,
%U 35010,2,9,63,345,1423,4459,11770,31607,95167,320589,2,10,80,514,2562,9820,30248,85634,258690
%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 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 = A300481(m,n)*exp(m) - a(m,n)*exp(-m).
%F a(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) * A080955(m,i) = Sum_{i=0..n} A127672(n,i) * A089258(i,m).
%e Array starts with:
%e m=0: 2, 1, 0, 3, 18, 95, 592, ...
%e m=1: 2, 2, 3, 10, 47, 256, 1610, ...
%e m=2: 2, 3, 8, 29, 130, 697, 4376, ...
%e m=3: 2, 4, 15, 66, 327, 1838, 11770, ...
%e m=4: 2, 5, 24, 127, 722, 4459, 30248, ...
%e ...
%o (PARI) { A300480(m,n) = if(n==0,return(2)); subst( serlaplace( 2*polchebyshev(n,1,(x+m)/2)), x, 1); }
%Y Values for m<=0 are given in A300481.
%Y Rows: A300482 (m=0), A300483 (m=1), A300484 (m=2), A300485 (m=-1), A102761 (m=-2).
%Y Columns: A007395 (n=0), A000027 (n=1), A005563 (n=2), A084380 (n=3).
%Y Cf. A000179 (almost row m=-2), A127672, A156995.
%K nonn,tabl
%O 0,1
%A _Max Alekseyev_, Mar 06 2018
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