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A300480 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.

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