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

0,1

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).

Links

Table of n, a(n) for n=0..63.

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).

Example

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, ...
...

Prog

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

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.

Sequence in context: A071572 A172176 A285930 * A342870 A143537 A125916

Adjacent sequences: A300477 A300478 A300479 * A300481 A300482 A300483

Keyword

nonn,tabl

Author

Max Alekseyev, Mar 06 2018

Status

approved