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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. 7
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 (list; table; graph; refs; listen; history; text; internal format)
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 * A143537 A125916 A283468

Adjacent sequences:  A300477 A300478 A300479 * A300481 A300482 A300483

KEYWORD

nonn,tabl

AUTHOR

Max Alekseyev, Mar 06 2018

STATUS

approved

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Last modified August 10 14:50 EDT 2020. Contains 336381 sequences. (Running on oeis4.)