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.

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, 592, 2, 7, 35, 127, 327, 697, 1610, 4277, 2, 8, 48, 218, 722, 1838, 4376, 11628, 35010, 2, 9, 63, 345, 1423, 4459, 11770, 31607, 95167, 320589, 2, 10, 80, 514, 2562, 9820, 30248, 85634, 258690

Offset: 0

Author

Max Alekseyev, Mar 06 2018

Keywords

• nonn
• tabl

Comments

a(m,n) is a polynomial in m of degree n.

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).

Examples

			Array starts with:
m=0: 2,  1,   0,    3,    18,     95,     592, ...
m=1: 2,  2,   3,   10,    47,    256,    1610, ...
m=2: 2,  3,   8,   29,   130,    697,    4376, ...
m=3: 2,  4,  15,   66,   327,   1838,   11770, ...
m=4: 2,  5,  24,  127,   722,   4459,   30248, ...
...
		

Crossrefs

Values for m<=0 are given in A300481.

Rows: A300482 (m=0), A300483 (m=1), A300484 (m=2), A300485 (m=-1), A102761 (m=-2).

Columns: A007395 (n=0), A000027 (n=1), A005563 (n=2), A084380 (n=3).

Cf. A000179 (almost row m=-2), A127672, A156995.

Programs

PARI

{ A300480(m,n) = if(n==0,return(2)); subst( serlaplace( 2*polchebyshev(n,1,(x+m)/2)), x, 1); }

Formula

a(m,n) = Sum_{i=0..n} A127672(n,i) * i! * Sum_{j=0..i} m^j/j!.

a(m,n) = Sum_{i=0..n} A127672(n,i) * A080955(m,i) = Sum_{i=0..n} A127672(n,i) * A089258(i,m).