A300481 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, 0, 0, 2, -1, -1, 3, 2, -2, 0, 2, 18, 2, -3, 3, 1, 7, 95, 2, -4, 8, -6, 2, 34, 592, 2, -5, 15, -25, 15, 13, 218, 4277, 2, -6, 24, -62, 82, -28, 80, 1574, 35010, 2, -7, 35, -123, 263, -269, 106, 579, 12879, 320589

Offset: 0

Author

Max Alekseyev, Mar 06 2018

Keywords

• sign
• tabl

Comments

Although negative values of m are not present here or in A300480, the two arrays are connected with the formula: a(m,n) = A300480(-m,n). Thus, they essentially represent two "halves" of the same array indexed by integers m.

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 = a(m,n)*exp(m) - A300480(m,n)*exp(-m).

Examples

			Array starts with:
m=0: 2,  1,  0,    3,   18,     95,    592, ...
m=1: 2,  0, -1,    2,    7,     34,    218, ...
m=2: 2, -1,  0,    1,    2,     13,     80, ...
m=3: 2, -2,  3,   -6,   15,    -28,    106, ...
m=4: 2, -3,  8,  -25,   82,   -269,    920, ...
...
		

Crossrefs

Values for m<=0 are given in A300480.

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

Columns (up to signs and offset): A007395 (n=0), A000027 (n=1), A005563 (n=2).

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

Programs

PARI

{ A300481(m,n) = A300480(-m,n); } \\ see code in A300480

Formula

a(m,n) = A300480(-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) * A292977(i,m).