A300482 -id:A300482 - cp's OEIS frontend

A300484 a(n) = 2 * Integral_{t>=0} T_n(t/2+1) * exp(-t) * dt, n>=0, where T_n(x) is n-th Chebyshev polynomial of first kind.

2, 3, 8, 29, 130, 697, 4376, 31607, 258690, 2368847, 24011832, 267025409, 3233119106, 42346123861, 596617706344, 8998126507307, 144651872924162, 2469279716419035, 44609768252582312, 850345380011532261, 17056474009400181122

Offset: 0

Max Alekseyev, Mar 06 2018

nonn

For any integer n>=0, 2 * Integral_{t=-2..2} T_n(t/2)*exp(-t)*dt = 4 * Integral_{z=-1..1} T_n(z)*exp(-2*z)*dz = A102761(n)*exp(2) - a(n)*exp(-2).

Row m=2 in A300480.
Row sums of A156995.
Cf. A102761, A300482, A300483, A300485.

{ A300484(n) = if(n==0, return(2)); subst( serlaplace( 2*polchebyshev(n, 1, (x+2)/2)), x, 1); }

a(n) = Sum_{i=0..n} A127672(n,i) * A010842(i).
a(n) = A300480(2,n) = A300481(-2,n).
a(n) = Sum_{m=0..n} A156995(n,m) = 2*n*Sum_{m=0..n} binomial(2*n-m, m)*(n-m)!/(2*n-m).

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.

Original entry on oeis.org

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

Max Alekseyev, Mar 06 2018

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).

			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, ...
...

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.

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

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).

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.

Original entry on oeis.org

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

Max Alekseyev, Mar 06 2018

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).

			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, ...
...

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.

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

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).

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.

Original entry on oeis.org

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

Max Alekseyev, Mar 06 2018

nonn

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).

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

Row m=1 in A300480.

Cf. A102761, A300482, A300484, A300485.

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

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 *)

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

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

A300485 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.

Original entry on oeis.org

2, 0, -1, 2, 7, 34, 218, 1574, 12879, 117938, 1195479, 13294412, 160967522, 2108289364, 29703846535, 447990339602, 7201792686815, 122938198060734, 2220989581865882, 42336203570931402, 849191837620701239, 17879821236086808098

Offset: 0

Max Alekseyev, Mar 06 2018

sign

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 = a(n)*exp(1) - A300483(n)*exp(-1).

Row m=1 in A300481.

Cf. A102761, A300482, A300483, A300484.

{ A300485(n) = if(n==0, return(2)); subst( serlaplace( 2*polchebyshev(n, 1, (x-1)/2)), x, 1); }

a(n) = Sum_{i=0..n} A127672(n,i) * A000166(i).

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

cp's OEIS Frontend

A300484 a(n) = 2 * Integral_{t>=0} T_n(t/2+1) * exp(-t) * dt, n>=0, where T_n(x) is n-th Chebyshev polynomial of first kind.

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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.

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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.

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Formula

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.

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Formula

A300485 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.

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Author

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Comments

Crossrefs

Programs

PARI

Formula

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.

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.