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.

2, 2, 3, 10, 47, 256, 1610, 11628, 95167, 871450, 8833459, 98233158, 1189398050, 15578268382, 219483388403, 3310225751098, 53214450175743, 908397242172212, 16411016615547530, 312824583201360248, 6274726126933368879, 132115002152296986730, 2913432246090160413827

Offset: 0

Author

Max Alekseyev, Mar 06 2018

Keywords

• nonn

Comments

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).

Links

• Robert Israel, Table of n, a(n) for n = 0..449

Crossrefs

Row m=1 in A300480.

Cf. A102761, A300482, A300484, A300485.

Programs

Maple

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

Mathematica

a[n_] := 2 Integrate[ChebyshevT[n, (t + 1)/2] Exp[-t], {t, 0, Infinity}];
Table[a[n], {n, 0, 22}] (* Jean-François Alcover, Feb 28 2019 *)

PARI

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

Formula

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

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

a(n) ~ exp(1) * n!. - Vaclav Kotesovec, Apr 15 2020