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A300483 a(n) = 2 * Integral_{t>=0} T_n((t+1)/2) * exp(-t) * dt, n>=0, where T_n(x) is n-th Chebyshev polynomial of first kind. 6

%I #22 Apr 15 2020 05:31:48

%S 2,2,3,10,47,256,1610,11628,95167,871450,8833459,98233158,1189398050,

%T 15578268382,219483388403,3310225751098,53214450175743,

%U 908397242172212,16411016615547530,312824583201360248,6274726126933368879,132115002152296986730,2913432246090160413827

%N a(n) = 2 * Integral_{t>=0} T_n((t+1)/2) * exp(-t) * dt, n>=0, where T_n(x) is n-th Chebyshev polynomial of first kind.

%C For any integer n>=0, 2 * Integral_{t=-1..1} T_n(t/2)*exp(-t)*dt = 4 * Integral_{z=-1/2..1/2} T_n(z)*exp(-2*z)*dz = A300485(n)*exp(1) - a(n)*exp(-1).

%H Robert Israel, <a href="/A300483/b300483.txt">Table of n, a(n) for n = 0..449</a>

%F a(n) = Sum_{i=0..n} A127672(n,i) * A000522(i).

%F a(n) = A300480(1,n) = A300481(-1,n).

%F a(n) ~ exp(1) * n!. - _Vaclav Kotesovec_, Apr 15 2020

%p seq(2*int(orthopoly[T](n,(t+1)/2)*exp(-t),t=0..infinity),n=0..50); # _Robert Israel_, Mar 06 2018

%t a[n_] := 2 Integrate[ChebyshevT[n, (t + 1)/2] Exp[-t], {t, 0, Infinity}];

%t Table[a[n], {n, 0, 22}] (* _Jean-François Alcover_, Feb 28 2019 *)

%o (PARI) { A300483(n) = if(n==0, return(2)); subst( serlaplace( 2*polchebyshev(n, 1, (x+1)/2)), x, 1); }

%Y Row m=1 in A300480.

%Y Cf. A102761, A300482, A300484, A300485.

%K nonn

%O 0,1

%A _Max Alekseyev_, Mar 06 2018

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Last modified March 28 20:05 EDT 2024. Contains 371254 sequences. (Running on oeis4.)